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ELEMENTS OF
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VECTOR ANALYSIS
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Arranged for the use of Students in Physics
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By J. WILLARD GIBBS,
Professor of Mathematical Physics in Yale College.
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NOT PUBLISHED.
NEW HAVEN:
Printed by Tuttle, Morehouse & Taylor
1881-4.
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ELEMENTS OF VECTOR ANALYSIS.
By J. Willard Gibbs.
[The fundamental principles of the following analysis are such as are familiar
under a slightly different form to students of quaternions. The manner in which
the subject is developed is somewhat different from that followed in treatises on.
quaternions, since the object of the writer does not require any use-of the con¬
ception of the quaternion, being simply to give a suitable notation for those rela¬
tions between vectors, or between vectors and scalars, which seem most import¬
ant, and which lend themselves most readily to analytical transformations, and
to explain some of theSe transformations. As a precedent for such a departure
from quaternionic usage, Clifford’s Kinematic may be cited. In this connection,
the name of Grassmann may also be mentioned, to whose system the following
method attaches itself in some respects more closely than to that of Hamilton.]
CHAPTER I.
COHCERNIHG THE ALGEBRA OF VECTORS.
Fundamental Notions.
1. Definition .—If anything lias magnitude and direction,
its magnitude and direction taken together constitute what is
called a vector.
The numerical description of a vector requires three num¬
bers, hut nothing prevents us from using a single letter for its
symbolical designation. An algebra or analytical method in
which a single letter or other expression is used to specify- a
vector may be called a vector algebra or vector analysis. -
Def . — As distinguished from vectors the real (positive or
negative) quantities of ordinary algebra are called scalars .*
As it is convenient that the form of the letter should indicate
whether a vector or a scalar is denoted, we shall use the small
* The imaginaries of ordinary algebra may be called biscalars, and that which
corresponds to them in the theory of vectors, bivectors. But we shall have no
occasion to consider either of these.
2
VECTOR ANALYSIS.
Greek letters to denote vectors, and the small English letters to
denote scalars. (The three letters, i, j, k, will make an excep¬
tion, to he jnentioned more particularly hereafter. Moreover,
7 r will he used in its usual scalar sense, to denote the ratio of
the circumference of a circle to its diameter.)
2. Def. —-Vectors are said to he equal when they are the
same both in direction and in magnitude. This equality is
denoted by the ordinary sign, as a—f The reader will ob¬
serve that this vector equation is the equivalent of three scalar
equations.
A vector is said to be equal to zero, when its magnitude is
zero. Such vectors may be set equal to one another, irrespec¬
tively of any considerations relating to direction.
3. Perhaps the most simple example of a vector is afforded
by a directed straight line, as the line drawn from A to B.
We may use the notation AB to denote this line as a vector,
i. e., to denote its length and direction without regard to its
position in other respects. The points A and B may be dis¬
tinguished as the origin and the terminus of the vector. Since
any magnitude may be represented by a length, any vector
may be represented by a directed line; and it will often be
convenient to use language relating to vectors, which refers to
them as thus represented.
Reversal of Direction , Scalar Multiplication and Division.
4. The negative sign (—) reverses the direction of a vector.
(Sometimes the sign -f may be used to call attention to the
fact that the vector has not the negative sign.)
Def. —A vector is said to be multiplied or divided by a
scalar when its magnitude is multiplied or divided by the
numerical value of the scalar and its direction is either un¬
changed or reversed according as the scalar is positive or nega¬
tive. These operations are represented by the same methods
as multiplication and division in algebra, and are to be regarded
as substantially identical with them. The terms scalar multi-
plication and scalar division are used to denote multiplication
and division by scalars, whether the quantity multiplied or
divided is a scalar or a vector.
5. Def. —A unit vector is a vector of which the magnitude
is unity.
Any vector may be regarded as the product of a positive
scalar (the magnitude of the vector) and a unit vector. .
The notation « 0 may be used to denote the magnitude of
the vector a.
VECTOR ANALYSIS.
3
Addition and Subtraction of Vectors.
6. Def.—r The sum of the vectors a, /?, Ac. (written a+p+
Ac.) is the vector found by the following process. Assuming
any point A, we determine successively the points B, C, Ac., so
that AB=a, BO=/9, Ac. The vector drawn from A to the
last point thus determined is the sum required. This is some¬
times called the geometrical sum, to distinguish it from an
algebraic sum or an arithmetical sum. It is also called the
resultant, and «, /?, Ac., are called the components. When the
vectors to be added are all parallel to the same straight line,
geometrical addition reduces to algebraic: when they have all
the same direction, geometrical addition like algebraic reduces
to arithmetical.
It may easily be shown that the value of a sum is not
affected by changing the order of two consecutive terms, and
therefore that it is not affected by any change in the order of
the terms. Again, it is evident from the definition that the
value of a sum is not altered by uniting any of its terms
in brackets, as a+[ft+f\+ Ac., which is in effect to substi¬
tute the sum of the terms enclosed for the terms themselves
among the vectors to be added. In other words, the commu¬
tative and associative principles of arithmetical and algebraic
addition hold true of geometrical addition.
7. Def. — A vector is said to be subtracted when it is added
after reversal of direction. This is indicated by the use of the
sign — instead of +.
8 . It is easily shown that the distributive principle of arith¬
metical and algebraic multiplication applies to the multiplica¬
tion of sums of vectors by scalars or sums of scalars : — i. e.,
(m + w + Ac.) [ar + /3+&c.]=ma + wa-(-&c.
+ mfi + n/3 + &c.
+ Ac.
9. Vector Equations. —If we have equations between sums
and differences of vectors, we may transpose terms in them,
multiply or divide by any scalar, and add or subtract the equa¬
tions, precisely as in the case of the equations of ordinary
algebra. Hence, if we have several such equations containing
known and unknown vectors, the processes of elimination and
reduction by which the unknown vectors may be expressed in
terms of the known are precisely the same, and subject to the
same limitations, as if the letters representing vectors repre¬
sented scalars. This will be evident if we consider that in the
multiplications incident to elimination in the supposed scalar
equations the multipliers are the coefficients of the unknown
quantities, or functions of these coefficients, and that such
4
VECTOR ANALYSIS.
multiplications may be applied to the vector equations, since
the coefficients are scalars.
10. Linear relation of four vectors , Coordinates. — If a, /?,
and y are any given vectors not parallel to the same plane, any
other vector p may he expressed in the form
p=aa + b/3+cy.
If a, /?, and y are unit vectors, a, b, and c are the ordinary
scalar components of p parallel to «, /9, and y. If y—OP,
(«, f y being unit vectors,) a, b, and c are the cartesian coordi¬
nates of the point P referred to axes through O parallel to
a , /9, and y. When the values of these scalars are given, p is
said to be given in terms of «, /2, and y. It is generally in this
way that the value of a vector is specified, viz., in terms of
three known vectors. For such purposes of reference, a sys¬
tem of three mutually perpendicular vectors have certain evi¬
dent advantages.
11 . Normal systems of unit vectors. — The letters i. j, k are
appropriated to the designation of a normal system of unit
vectors , i. e., three unit vectors, each of'which is at right angles
to the other two and determined in direction by them in a
perfectly definite manner. We shall always suppose that k is
on the side of the i-j plane on which a rotation from i to j
(through one right angle) appears counter-clock-wise. In other
words, the directions of i, j, and k are to be so determined
that if they be turned (remaining rigidly connected with each
other) so that i points to the east, and j to the north, k will
point upward. When rectangular axes of X, Y, and Z are
employed, their directions will be conformed to a similar con¬
dition, and i, j , k (when the contrary is not stated) will be
supposed parallel to these axes respectively. We may have
occasion to use more than one such system of unit vectors,
just as we may use more than one system of coordinate axes.
In such cases, the different systems may be distinguished by
accents or otherwise.
12. Numerical computation of a geometrical sum. —If
p —aa + b/3 + cy,
< 3 —ci r a + b'fi + c'y,
&c.,
then
p-p g -j- &c. — {a -f- a' + &c.j cx.-\- {b -1 - b +&c.)/?+(c + c +&c i)y.
I. e., the coefficients by which a geometrical sum is expressed
in terms of three vectors are the sums of the coefficients by
which the separate terms of the geometrical sum are expressed
in terms of the same three vectors.
VECTOR ANALYSIS.
5
Direct and Shew Products of Vectors.
13. Def —The direct product of a and ft (written a. ft) is the
scalar quantity obtained by multiplying the product of their
magnitudes, by the cosine of the angle made by their direc¬
tions.
14. Def .—The skew product of a and ft (written axft) is a
vector function of a. and ft. Its magnitude is obtained by
multiplying the product of the magnitudes of a. and /3 by the
sine of the angle made by their directions. Its direction is at
right angles to a. and ft, and on that side of the plane contain¬
ing a and ft (supposed drawn from a common origin), on which
a rotation from o .: to ft through an arc of less than 180° appears
counter-clock-wise.
The direction of aXft may also be defined as that in which
an ordinary screw advances as it turns so as to carry a : toward ft.
Again, if a be directed toward the east, and ft lie in the
same horizontal plane and on the north side of a, aX ft will be
directed upward.
15. It is evident from the preceding definitions that
a.13=13.a, and aX [3—— [3Xoi.
16. Moreover,
[gn]. (3 = a. [n/3] = n[a. /3 ],
and [na]x/3—aX[nf3]=n[aXf3].
The brackets may therefore be omitted in such expressions.
17. From the definitions of Mo. 11 it appears that
i. i —j.j—k .k= 1,
i.j =j. i=i. k — k. i =j. k=k.j = 0,
iXi= 0, jXj= 0, kxk= 0,
i Xj=k, jXk=i , k X i =j,
jXi——k , k xj= — q i X, k= —j.
18. If we resolve ft into two components ft' and ft", of which
the first is parallel and the second perpendicular to a, we shall
have
a.j3 = a.f3’ and aX f3=ot.X j3".
19. a.[/3 + y] = a./3+a.y and aX[/3 + y] = aX/3 + aXy.
To prove this, let a—ft+y, and resolve each of the vec¬
tors ft, y, a into two components, one parallel and the other
perpendicular to a. Let these be ft', ft", y', y", a', a". Then
the equations to be proved will reduce by the last section to
a . a'—a . ft + a . y' and <xXa"=aX f3" + aXy".
6
VECTOR ANALYSIS.
Now since a=ap s jT we may form a triangle in space, the sides
of which shall be ft, y, and a. Projecting this on a plane per¬
pendicular to a, we obtain a triangle having the sides ft", y",
and a", which affords the relation a"—ft !, -\-y". If we pass
planes perpendicnlar to a through the vertices of the first
triangle, they will give on a line parallel to a segments eqnal
to ft',y',a\ Thus we obtain the relation a'=ft ; +y ; . There¬
fore a. 0 '=a.ft'+a.y', since all the cosines involved in these pro¬
ducts are equal to unity. Moreover, if a is a unit vector, we
shall evidently have a X a"= aX ft" + a X y", since the effect of
the skew multiplication by a upon vectors in a plane perpen¬
dicular to a is simply to rotate them all 90° in that plane. But
any case may be reduced to this by dividing both sides of the
equation to he proved by the magnitude of a. The proposi¬
tions are therefore proved.
20. Hence,
[n + /t] . y = a.y + ft.y, [a + ft] X y=aXy + ftX y,
[a' + /T] . {y-\-d^] = a.y-^(.d~\-ft.y + ft.6,
[a + /3\x[y+$]=aXy + a^S + fiXy + fiXd;
and, in general, direct and skew products of sums of vectors
may be expanded precisely as the products of sums in algebra,
except that in skew products the order of the factors must not
he changed without compensation in the sign of the term. If
any of the terms in the factors have negative signs, the signs
of the expanded product (when there is no change in the order
of the factors), will he determined by the same rules as in
algebra. It is on account of this analogy with algebraic prod¬
ucts that these functions of vectors are called products and
that other terms relating to multiplication are applied to them.
21. Numerical, calculation of direct and skew products —
The properties demonstrated in the last two paragraphs (which
may be briefly expressed by saying that the operations of
direct and skew multiplication are distributive) afford the rule
for the numerical calculation of a direct product, or of the
components of a skew product, when the rectangular compo¬
nents of the factors are given numerically. In fact,
if a—xi + yj+zk , and fi=x r i+y'j+ztk ;
a. ft—xx' + yy' + zz r ,
and ax ft={yd—zy')i + (zx'—xz')j + (xy' — yx')k.
22 . Representation of the area of a parallelogram by a
skew products. —It will be easily seen that a X ft represents in
magnitude the area of the parallelogram of which a and ft (sup¬
posed drawn from a common origin) are the sides, and that it
represents in direction the normal to the plane of the parallel-
VECTOR ANALYSIS.
1
ogram on the side on which the rotation from a toward
appears counter-clock-wise.
23. Representation of the volume of a parallelopiped by a
triple product .—It will also be seen that aXfi.f represents
in numerical yalue the volume of the parallelopiped of which
a, /9, and y (supposed drawn from a common origin), are the
edges, and that the value of the expression is positive or nega¬
tive according as y lies on the side of the plane of a and /9 on
which the rotation from a. to /9 appears counter-clock-wise, or
on the opposite side.
24. Hence,
aX/3.y=/3Xy.a=yXaf=y-aX/3=a./3xy
—fi.yXot— —ft X a.y= — y X fi.a= — a X y.fi
= —y.pXa=—a.yXft=—fi.aXy.
It will he observed that all the products of this type, which
can he made with three given vectors, are the same in numer¬
ical value, and that any two such products are of the same or
opposite character in respect to sign, according as the cyclic
order of the letters is the same or different. The product van¬
ishes when two of the vectors are parallel to the same line, or
when the three are parallel to the same plane.
This kind of product may be called the scalar product of the
three vectors. There are two other kinds of products of three
vectors, both of which are vectors, viz: products of the type
( a.ffy or y (a . /9), and products of the type ax{fXy] or
\jXff\Xa.
25. i.jX k—j.kx i—k.iXj= 1. i.kxj—k.j X i=jiXk= — 1.
' From these equations, which follow immediately from those of
No. 17, the propositions of the last section might have been
derived, viz: by substituting for a, /9, and y, respectively,
expressions of the form xi+yj+zk, x'i+y'j+z'k, and x"i~\-y"j
+z"kf Such a method, which may be called expansion in
terms of i, j, and k, will on many occasions afford very simple,
although perhaps lengthy, demonstrations.
26. Triple products containing only two different letters .—
The significance and the relations of (a. a)/9, (a.fa, and
aX [«X/9] will be most evident, if we consider /9 as made up of
* Since the sign x is only used between vectors, the skew multiplication in
expressions of this kind is evidently to be pel-formed first. In other words, the
above expression must be interpreted as [a x /?] . y.
f The student who is familiar with the nature of determinants will not fail to
observe that the triple product a.f3 x y is the determinant formed by the nine
rectangular components of a, (3, and y, nor that the rectangular components of
a x (3 are determinants of the second order formed from the components of a and
f3. (See the last equation of No. 21.)
8
VECTOR ANALYSIS.
two components, f and B'\ respectively parallel and perpen¬
dicular to a. Then
fi—fi'f- (3",
( a. (3) a=[a.(3') a=[a.a) (3\
aX[&X fi] = ax[ax /3 r '] = — (a. a) (3".
Hence, ffX [a X (3~\=d[a.(3) a—(a.a) (3.
27. General relation of the vector products of three factors .
—In the triple product « X [/? X y] we may set
a=l/3 + my + n/3xy ,
unless /9 and y have the same direction. Then
ax[/3xy]=lfix[/3xy]+myx[/3xy]
—l {fi.y) f3—l(/3./3) y—m (y.fi) y + m (y.y) (3
■=(l/3.y+my.y) )3 — (l/3.(3 + my.f3 ) y.
But lf3.y + my.y=a.y, and 1(3.(3my. (3—a. (3.
VoiVjly -y.nfr Therefore ax[(3Xy] = (a.y) (3— [a.(3) y,
which is evidently true, when {3 and y have the same directions.
It may also be written
[yX/3] Xa=P (y.a) — y {(3.a).
28. This principle may he used in the transformation of
more complex products. It will he observed that its applica¬
tion will always simultaneously eliminate, or introduce, two
signs of skew multiplication.
The student will easily prove the following identical equa¬
tions, which, although of considerable importance, are here
given principally as exercises in the application of the preced¬
ing formulae.
29. aX[/3xy]+/3X[yX<x} + yX[aX/3]= 0.
30. [ax(3].[yX6] = (a.y) (/3.6) — (a.6) (/lyf
31. [aX/3]X[yXS]—(a.yXd) (3— {fi.yx 6) a
= (a./3x<3) y—{a.(3xy) d.
s
32. axipX[yXd]]=(a.yXd) (3—[a.(3) yX6
— [(3.6) axy—{/3.y ) «Xd.
33. [aX/3]. [yX6]x[£XZ]=(a-ftx6){y.£XZ)—(a./3xy)(6.ex£)
= (a.f3x Z) fy X 6)-\-[a.(3x e) (<g. y X d)
= {y.dxa)[(3.exd) — {y-6x(3)(a.exd).
34. [aXft]. \_(3Xy\X\yXoi\ — (a.f3XyY'
11.
i avfhy -4- Vy v&p ~ &
3d.
&-y. — oiS.fiy
y. —
3l.
W<ip>\/Y$ — ^ \/y§ cs
3*.
VvVfiVy? = /3.0tK r <f_ XuV/iy
~ V<*y
— fy.
33.
v «fiVY f 2Vzt ) ~ < V(iS'.- r Ve$ -
C iV/iy. JVcJ
V
Voi/iV Y/i r V r u =
YECTOE ANALYSIS.
9
35. The student will also easily convince himself that a
product formed of any number of letters (representing vectors)
combined in any possible way by scalar, direct, and skew mul¬
tiplications may be reduced by the principles of Nos. 24 and
27 to a sum of products, each of which consists of scalar fac¬
tors of the forms a. ft and a.ftxy, with a single vector factor
of the form a or a X ft, when the original product is a vector.
36. Elimination of scalars from rector eolations . — It has
already been observed that the elimination of vectors from
ecpiations of the form
aa + bfi + cy + d S + &c. = 0
is performed by the same rule as the eliminations of ordinary
algebra. (See No. 9.) But the elimination of scalars from
such equations is at least formally different. Since a single
vector equation is the equivalent of three scalar equations, we
must be able to deduce from such an equation a scalar equa¬
tion from which two of the scalars which appear in the orig¬
inal vector equation have been eliminated. We shall see how
this may be clone, if we consider the scalar equation
a a. X + b/3. X + cy. X + cl 6. X + &c. == 0,
which is derived from the above vector equation by direct mul¬
tiplication by a vector X. We may regard the original equa¬
tion as the equivalent of the three scalar equations obtained by
substituting for a, ft, y, d, etc., their X-, Y-, and Z- compo¬
nents. The second equation would be derived from these by
multiplying them respectively by the X-, Y-, and Z- compo¬
nents of X and adding. Hence the second equation may be
regarded as the most general form of a scalar equation of the
first degree in a, b, c, d, etc., which can be derived from the
original vector equation or its equivalent three scalar equations.
If we wish to have two of the scalars, as b and c, disappear, we
have only to choose for X a vector perpendicular to ft and y.
Such a vector is ftXy. We thus obtain
aa.fi Xy+ d6.fi x y 4- &c.=0.
37. Relations of four rectors. —By this method of elimina¬
tion we may find the values of the coefficients a, b, and c in
the equation
p=aa-\-bfi + cy, (1)
by which any vector p is expressed in terms of three others.
(See No. 10.) If we multiply directly by ft X y, y X a, and
axft, we obtain
p.fiXy=aa.fixy, p.yXa=bfi.yXcv, p.aX fi—cy.aX fi ; (2)
whence
2
10
VECTOR, ANALYSIS.
a=M>< T; b= RZl If P-<*XP
at.fiXy 9 a.ftxy’ a.fixy ^ '
By substitution of these values, we obtain the identical equa¬
tion,
(a.fixy) p=(p.fixy) a + (p.yx<x) fi+(p.axfi) y ( 4 )
(Compare Ho. 31.) If we wish the four vectors to appear
symmetrically in the equation we may write
{o'.fSxy) p—{/3.yXp) a + (y.pXa) /3—(p.axW ) y = 0. ( 5 )
If we wish to express p as a sum of vectors having directions
perpendicular to the planes of a and ft, of ft and y, and of y and
a, we may write
P=efiXy+fyXa+gax/3. ( 6 )
To obtain the values of e,f g, we multiply directly by a, by ft,
and by y. This gives
e — _ P’ a f— _ P -fi r/= _ p -y
ft.yXoi ' y.aXfP J at.fiXy
(1)
Substituting these values we obtain the identical equation
(a.ftXy) p={p.a) fiXy+(p.fi) yX<x+(p.y) acXfi. (8)
(Compare Ho. 32.)
38. Reciprocal systems of rectors .—The results of the pre¬
ceding section may be more compactly expressed if we use the
abbreviations
a ./_ fixy' yxa axfi
ac.fi Xy* fi.yXoi ' y.aXfi'
0)
The identical equations (4) and (8) of the preceding number
thus become
p=(p.a) a+(p.fi') fi+(p.y') y,
p=(p.a) a'+(p.fi ) fi'+(p.y) y'.
(2))
(3)
We may infer from the similarity of these equations that the
relations of «, ft, y, and o!, ft', y' are reciprocal; a proposition
which is easily proved directly. For the equations
_ fi'xy o y'Xoc' _ a'xfi’
a ~a’.p'xy” 1 P'.y'Xa” r ~y',a'xP'
are satisfied identically by the substitution of the values of
o!, ft ', and y' given in equations (1). (See Hos. 31 and 34.)
Ref .—It will be convenient to use the term reciprocal to
designate these relations, i. e., we shall say that three vectors
are reciprocals of three others, when they satisfy relations sim¬
ilar to those expressed in equations (1) or (4).
ph
aV/4,
VAC
rvpy
r "
I
ary'* I
VECTOR ANALYSIS.
11
With this understanding we may say:—
The coefficients hy which any vector is expressed in terms of
three other vectors are the direct products of that vector with
the reciprocals of the three.
Among other relations which are satisfied hy reciprocal sys¬
tems of vectors are the following:
a.a'—fi.fi’ — y.y' z=l\.
[a.fixy) =
•(See No. 34.)
aXot' + fiXfi' + y Xy'=o.
(See No. 29.)
A system of three mutually perpendicular unit
reciprocal to itself, and only such a system.
The identical equation
( 5 ) =/
(»)
vectors is s ^
P= (P-i) i + (P-j) j+ (p.Je) k
may be regarded as a particular case of equation (2).
The system reciprocal to a.X[ 3, fi X y, yXa is
a
fi
fi a. fix y* oc.fi xy' -a.fixy
t
( 8 ) ?= L.if+j.ptrU^
Vfa.
39. Scalar equations of the first degree with respect to
unknown rector. —It is easily shown that any scalar equation
of the first degree with respect to an unknown vector p 9 in
which all the other quantities are known, may he reduced to
the form
p.ac=za,
(Ayi Y,y
Y al _ y
tW aA
o an / #
in which a. and a are known. (See ISTo. 35.) Three such
equations will afford the value of p (hy equation (8) of No. 37,
or equation (3) of No. 38), which may he used to eliminate p
from any other equation either scalar or vector.
When we have four scalar equations of the first degree with
respect to p, the elimination may he performed most symmet¬
rically hy substituting the values of p.a etc., in the equation,
(p.oc) (fi.yX 8 ) — (p.fi) (y.Sxoc)+ (p.y) (d.ax fi) — (p.d) ( a.fixy ),
which is obtained from equation (8) of No. 37 hy multiplying
directly hy d. It may also be obtained from equation (5) of No.
37 hy writing a for p, and then multiplying directly by p.
40. Solution of a rector equation of the first degree with
respect to the unknown rector .—It is now easy to solve an
equation of the form
8—a (k.p) + fi (p.p) + y (v.p),
~ c4 - Xy -y f 2 "- Yjfi
~ +-/5-r + ^.' ? )p
f - ««'+ 0+ v.y) S'
n.fi +. vqq +p.f- rfy-v)
( 1 )
err-
2 = fp
i
12
VECTOR ANALYSIS.
where «, /9, y, d , A, p, and v represent known vectors. Multi¬
plying directly by /?Xp, by yX«, and by axft we obtain
fi.yX8=(f3.yXoi) (A.p), yaX 8=(y.aXfi) (p.p),
a.fix8 = {a.fixy) (u.p);
or a.6—\.p, fi'.d—jj.p, y'.8=y.p,
where a 1 , f, y' are the reciprocals of a, j3, y. Substituting
these values in the identical equation
p—A'/A.p) 4 - p'(p.p) 4- r'(y.p ),
in which A', p', v' are the reciprocals of A, p, v, (see Ho. 38,)
we have
p=V(a |d) + p/(/3'.d) 4 (2)
which is the solution required.
It results from the principle stated in Ho. 35, that any
vector equation of the first degree with respect to p may be
reduced to the form
8 = p(A.p) + [8 (p.p) 4- y{y.p) + ap 4 s x p.
But ap—oK! (A.p) 4«p'(p.p) + ar'(r.p ),
and £x p=£X A'(A.p) + fX p'(p-p) + sX y'(v.p ),
a.(XV+-) P where A', //, i/ represent, as before, the reciprocals of A, p, v.
f By substitution of these values the equation is reduced to the
- Vq\ . A p form of equation (1), which may therefore be regarded as the
4-..« * most general form of a vector equation of the first degree with
respect to p.
41. Relations between two normal systems of unit rectors .—
If i, f A, and i/, k! are two normal systems of unit vectors,
we have
a — (i.i f )i 4 {j.i' )j+ ( Je.i ' )k, j
f = (if )i + 07 )j+ ( k.f )k, | ( 1 )
k'=(i.k')i + (j.k')j 4 (k.k')k, )
and
i = ( i.i ) i' + ( ij’lf + ( l/. J ')/■:, i
j=(j.iy + (jj')j' + (j.k')k',\ (2)
k = (k.i')% 4 (k.j )j' 4 ( k.k')k'. )
(See equation 8 of Ho. 38.)
The nine coefficients in these equations are evidently the
cosines of the nine angles made by a vector of one system with
a vector of the other system. The principal relations of these
cosines are easily deduced. By direct multiplication of each
of the preceding equations with itself, we obtain six equations
of the type
(i.i'Y + (j.ir + (k.ir=l- ( 3 )
4- 4.
VECTOR ANALYSIS.
13
By direct multiplication of equations (1) with each other, and
of equations (2) with each other, we obtain six of the type
(W) (if) + (j.i') (j.jj + (hi') (hf) = 0. (4)
By skew multiplication of equations (1) with each other, we
obtain three of the type
k'={(j.ij (hf)-(hi') I (hi') (i.j')-(i.i') hjj }j
+ {(bb) (j-j')-(j-i') (ij')}h
Comparing these three equations with the original three, we
obtain nine of the type
iM=(j.i')(hf)-(hi')(jf). (5)
Finally, if we equate the scalar product of the three right hand
members of (1) with that of the three left hand members, we
obtain
(U') 07) (hh) + (if) (j.h) (hi') + (i.h) (j.i') (kf)
-(hi') 07) (i.A/)-(V) tw (M)-(hh) (j.i') 07)=1.
( 6 )
Equations (1) and (2) (if the expressions in the parentheses
are supposed replaced by numerical values) represent the linear
relations which subsist between one vector of one system and
the three vectors of the other system. If we desire to express
the similar relations which subsist between two vectors of one
system and two of the other, we may take the skew products
of equations (1) with equations (2), after transposing all terms
in the latter. This will afford nine equations of the type
(if)h-(i.h)f=(hi')j-(j.i')h ( 7 )
14
VECTOR ANALYSIS.
CHAPTER II.
CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS
OF VECTORS.
42. Differentials of vectors .—The differential of a vector is
the geometrical difference of two values of that vector which
differ infinitely little. It is itself a vector, and may make any
angle with the vector differentiated. It is expressed by the same
sign iff) as the differentials of ordinary analysis.
With reference to any fixed axes, the components of the
differential of a vector are manifestly equal to the differentials
of the components of the vector, i. e., if «, /3, and y are fixed
unit vectors, and
p=xa + yfi+zy,
dp=dx a J r dyfj + dz y.
43. Differential of a function of several variables .—The
differential of a vector or scalar function of any number of
vector or scalar variables is evidently the sum (geometrical or
algebraic, according as the f unction is vector or scalar,) of the
differentials of the function due to the separate variation of
the several variables.
44. Differential of a product .—The differential of a product
of any kind due to the variation of a single factor is obtained
by prefixing the sign of differentiation to that factor in the
product. This is evidently true of differentials, since it will
hold true even of finite differences.
45. From these principles we obtain the following identical
equations:
d[a -f- fp —^ da -|- dfi^ (l)
d(na)=dn a + n da , (2)
d(a. j3)— da. ft-y a.dfi, (3)
d[aX fi]=daX +aXdfi, (4)
d(a.fix y)= da.fi Xy + a. dp Xy + a.ft X dy, (5)
d[(a.fi)y] = (da.fi)y+ (a.dfi)y + (a.fi)dy. (6)
46. Differential coefficient with respect to a scalar .—The
quotient obtained by dividing the differential of a vector due
to the variation of any scalar of which it is a function by the
differential of that scalar is called the differential coefficient of
the vector with respect to the scalar, and is indicated in the
same manner as the differential coefficients of ordinary analysis.
VECTOR ANALYSIS.
15
If we suppose the quantities occurring in the six equations
of the last section to he functions of a scalar t, we may substi¬
tute — for d in those equations since this is only to divide all
Cllr
terms by the scalar dt.
41. Successive differentiations .—The differential coefficient
of a vector with respect to a scalar is of course a finite vector,
of which we may take the differential, or the differential coef¬
ficient with respect to the same or any other scalar. We thus
obtain differential coefficients of the higher orders, which are
indicated as in the scalar calculus.
A few examples will serve for illustration.
If p is the vector drawn from a fixed origin to a moving
point at any time t, y will he the vector representing the
(JjTj
ddp
velocity of the point, and the vector representing its accel-
(lu
eration.
If p is the vector drawn from a fixed origin to any point on
a curve, and s the distance of that point measured on the
(7 Q
curve from any fixed point, is a unit vector, tangent to the
CtS
• ^ Q
curve and having the direction in which s increases: —- is a
ds
vector directed from a point on the curve to the center of curv¬
ature, and equal to the curvature: ~ X is the normal to the
ds ds
osculating plane, directed to the side on which the curve appears
described counter-clock-wise about the center of curvature,
and equal to the curvature. The tortuosity (or rate of rotation
of the osculating plane, considered as positive when the rota¬
tion appears counter-clock-wise as seen from the direction in
which s increases,) is represented by
dp d 3 p d 3 p
ds * ds 2 ^ ds 3
dp df>
ds' 1 -* ds ^
48. Integration of an equation between differentials .—If t
and u are two single-valued continuous scalar functions of any
number of scalar or vector variables, and
then
dt—du ,
t — u -f- a.
where a is a scalar constant.
16
VECTOR ANALYSIS.
Or, if t and co are two single-valued continuous vector func¬
tions of any number of scalar or vector variables, and
dr—dcsD ,
then t=go+o',
where a is a vector constant.
"When the above hypotheses are not satisfied in general, but
will be satisfied if the variations of the independent variables
are confined within certain limits, then the conclusions will
hold within those limits, provided that we can pass by continu¬
ous variation of the independent variables from any values
within the limits to any other values within them, without
transgressing the limits.
49. So far, it will be observed, all operations have been
entirely analogous to those of the ordinary calculus.
Functions of Position in Space.
50. Pef. —If u is any scalar function of position in space,
(i. e., any scalar quantity having continuously varying values
in space,) pu is the vector function of position in space which
has everywhere the direction of the most rapid increase of u,
and a magnitude equal to the rate of that increase per unit of
length, pu may be called the derivative of u, and u, the
primitive of pu.
We may also take any one of the bTos. 51, 52, 53 for the
definition of pu.
51. If p is the vector defining the position of a point in space,
du—pu.dp.
52.
53.
.du .du 7 du
r u=l 7x +J 7j +k
dz
du
dx
—i.pu ,
du . du 7
p z =lc F u -
dy
54 . Def. —If co is a vector having continuously varying
values in space,
. doo . dco dco ,
< 7 - 6,= f* + ^ +i -&’ (1)
. da) . da) dco
and f 7x<a =* x ^ + ' ?x ^ +/l ' x &-
p.co is called the divergence of co and p X co its curl.
If we set
rc)=Xi + Yj +Z/q
VECTOB ANALYSIS.
11
we obtain by substitution tbe equations
dX d,Y dZ
00 dx dy ^ dz'
, •/
and /7 X go— ^(
V
dZ
dy
dX\
dyr
wbicli may also be regarded as defining p.co and y X co.
55. Surface-integrals .—The integral ffco.da , in which da
represents an element of some surface, is called the surface-
integral of co for that surface. It is understood here and else¬
where, when a vector is said to represent a plane surface, (or
an element of surface, which may be regarded as plane,) that
the magnitude of the vector represents the area of the surface,
and that the direction of the vector represents that of the nor¬
mal drawn toward the positive side of the surface. When the
surface is defined as the boundary of a certain space, the out¬
side of the 'surface is regarded as positive.
The surface-integral of any given space (i. e., the surface-
integral of the surface bounding that space) is evidently equal
to the sum of the surface-integrals of all the parts into which
the original space may be divided. Tor the integrals relating
to the surfaces dividing the parts will evidently cancel in such
a sum.
The surface-integral of to for a closed surface bounding a
space dr infinitely small in all its dimensions is
J7. GO dv.
This follows immediately from the definition of yco, when
the space is a parallelopiped bounded by planes perpendicular
to i, j, k. In other cases, we may imagine the space—-or rather
a space nearly coincident with the given space and of the same
volume dv —to be divided up into such parallelopipeds. The
surface-integral for the space made up of the parallelopipeds
will be the sum of the surface-integrals of all the parallelo¬
pipeds, and will therefore be expressed by p.co dv. The sur¬
face-integral of the original space will have sensibly the same
value, and will therefore be represented by the same formula.
It follows that the value of p.co does not depend upon the
system of unit vectors employed in its definition.
It is possible to attribute such a physical signification to
the quantities concerned in the above proposition, as shall
make it evident almost without demonstration. Let us suppose
co to represent a flux of any substance. The rate of decrease
of the density of that substance at any point will be obtained
by dividing the surface-integral of the flux for any infinitely
small closed surface about the point by the volume enclosed,
3
18
VECTOR ANALYSIS.
This quotient must therefore he independent of the form of
the surface. We may define p.ta as representing that quotient,
and then obtain equation (1) of bio. 54 by applying the general
principle to the case of the rectangular parallelopiped.
56. Shew surface-integrals. —The integral fjdaXto maybe
called the skew surface-integral of to. It is evidently a vector.
Tor a closed surface hounding a space dr infinitely small in all
dimensions, this integral reduces to pXtodv, as is easily shown
by reasoning like that of bio. 55.
51. Integration .—If dr represents an element of any space,
and da an element of the hounding surface,
ff/V • dv =ff a) - d <?•
For the first member of this equation represents the sum of the
surface integrals of all the elements of the given space. We
may regard this principle as affording a means of integration,
since we may use it to reduce a triple integral (of a certain
form) to a double integral.
The principle may also he expressed as follows:
The surface-integral of any vector function of position in
space for a closed surface is equal to the volume-integral of
the divergence of that function for the space enclosed.
58. Line-integrals .—The integral J'to.dp, in which dp de¬
notes the element of a line, is called the line-integral of to for
that line. It is implied that one of the directions of the line
is distinguished as positive. When the line is regarded as
hounding a surface, that side of the surface will always he
regarded as positive, on which the surface appears to he cir¬
cumscribed counter-clock-wi se.
59. Integration. —From JNo. 51 we obtain directly
fyu .dp—y!'—u
where the single and double accents distinguish the values
relating to the beginning and end of the line.
In other words,—The line-integral of the derivative of any
(continuous) scalar function of position in space is equal to the
difference of the values of the function at the extremities of
the line. For a closed line the integral vanishes.
60. Integration. —The following principle may he used to
reduce double integrals of a certain form to simple integrals.
If da represents an element of any surface, and dp an
element of the hounding line,
ffV X oo.d,(i—f oo.dp.
In other words,—The line-integral of any vector function of
position in space for a closed line is equal to the surface-inte-
VECTOR ANALYSIS.
19
gral of the curl of that function for any surface hounded by
the line.
To prove this principle, we will consider the variation of the
line-integral which is due to a variation in the closed line for
which the integral is taken. We have, in the first place,
6/oo.dp=/S oo.dp+f go. 6 dp.
But QD.ddp—d{GD.S p)—doo.d p.
Therefore, since fdico.dp)— 0 for a closed line,
Now
and
6foo.dp=-fdGD.dp—fdGD.d p.
1
8
1
doo
|_*d /J) j
doo=J>
cl GO ,
dx
dx
= 2
'doo
dx {t - dp) ’
where the summation relates to the coordinate axes and con¬
nected quantities. Substituting these values in the preceding
equation, we get
Sf,-W.dp=f2((i.dp) - (Up)(^. Sp^j,
or by Ho. 30,
5/GD.dp—f2
d(sD
dx
. [dp X dp] =fp X go. [dp X dp].
But dpxdp represents an element of the surface generated by
the motion of the element dp, and the last member of the
equation is the surface-integral of j 7 Xco for the infinitesimal
surface generated by the motion of the whole line. Hence,
if we conceive of a closed curve passing gradually from an
infinitesimal loop to any finite form, the differential of the line-
integral of co for that curve will be equal to the differential of
the surface integral of y X co for the surface generated: therefore,
since both integrals commence with the value zero, they must
always be equal to each other. Such a mode of generation
will evidently apply to any surface closing any loop.
61. The line-integral of co for a closed fine bounding a plane
surface da infinitely small in all its dimensions is therefore
j7 X go. da.
This principle affords a definition of pXco which is inde¬
pendent of any reference to coordinate axes. If we imagine
a circle described about a fixed point to vary its orientation
while keeping the same size, there will be a certain position of
the circle for which the line-integral of co will be a maximum,
unless the line-integral vanishes for all positions of the circle.
The axis of the circle in this position, drawn toward the side
20
VECTOR ANALYSIS.
on which a positi ve motion in the circle appears counter-clock¬
wise, gives the direction of y X co, and the quotient of the inte¬
gral divided by the area of the circle gives the magnitude of
/7 X co.
y, y., and (7 X allied to Functions of Functions of Position.
62. A constant scalar factor after y, y., or y x may be
placed before the symbol.
63. If f(u) denotes any scalar function of u, and fO l) the
derived function,
Cf(f)=f {u)yu.
64. If u or co is a function of several scalar or vector varia¬
bles, which are themselves functions of the position of a single
point, the value of yu or y.co or yXco will be equal to the sum
of the values obtained by making successively all but each one
of these variables constant.
65. By the use of this principle, we easily derive the follow¬
ing identical equations:
y(t + U ) = yt+ yU. (1)
y.(r + oo) = y.r + y.Go. yX[r + co] = yXr + yXGo. (2)
y(tu)=uyt + tyv. (3)
y.(uGo) = CsD.yu-\-v,y.Go. (4)
y X \u go] = ny Xgo—goX yu. (5)
y.\rXGo] — co.yX'r—r.yXco. (6)
The student will observe an analogy between these equations
and the formula} of multiplication. (In the last four equations
the analogy appears most distinctly when we regard all the fac¬
tors but one as constant.) Some of the more curious features
of this analogy are due to the fact that the y contains implic¬
itly the vectors i, j, and k, which are to be multiplied into
the following quantities.
Combinations of the Operators y, y., and y X .
66. If u is any scalar function of position in space,
yxyu= o,
as may be derived directly from the definitions of these ope¬
rators.
6T. Conversely, if co is such a vector function of position in
space that
yXGO — 0,
VECTOR ANALYSIS.
21
co is the derivative of a scalar function of position in space.
This will appear from the following considerations:
The line-integral fco.dp will vanish for any closed line, since
it may he expressed as the surface-integral of p X co. (No. 60.)
The line-integral taken from one given point P / to another
given point P 77 is independent of the line between the points
for which the integral is taken. (For, if two lines joining the
same points gave different values, by reversing one we should
obtain a closed line for which the integral would not vanish.)
If we set u equal to this line-integral, supposing P 77 to be
variable "and P 7 to be constant in position, u will be a scalar
function of the position of the point P 77 , satisfying the condi¬
tion du=co.dp, or, by No. 51, pu—co. There will evidently
be an infinite number of functions satisfying this condition,
which will differ from one. another by constant quantities.
If the region for which pXco — 0 is unlimited, these func¬
tions will be single-valued. If the region is limited, but
acyclic,* the functions will still be single-valued and satisfy
the condition pu=co within the same region. If the region is
cyclic, we may determine functions satisfying the condition
p'u—co within the region, but they will not necessarily be
single-valued.
68. If co is any vector function of position in space,
p.pXco= 0. This may be deduced directly from the defini¬
tions of No. 54.
The converse of this proposition will be proved hereafter.
69. If u is any scalar function of position in space, we have
by ISTos. 52 and 54
(d* d 2 cV\
r-r v= W + ty + **r
TO. Def . — If co is any vector function of position in space,
we may define p.pco by the equation
/ cf d* cl 2 \
^= 1 *XThf + dsN
* If every closed line within a given region can contract to a single* point
without breaking its continuity, or passing out of the region, the region is called
acyclic, otherwise cyclic.
A cyclic region may be made acjmlic by diaphragms, which must then be re¬
garded as forming part of the surface bounding the region, each diaphragm
contributing its own area twice to that surface. This process may be used to
reduce many-valued functions of position in space, having single-valued deriva¬
tives, to single-valued functions.
When functions are mentioned or implied in the notation, the reader will always
understand single-valued functions, unless the contrary is distinctly intimated, or
the case is one in which the distinction is obviously immaterial. Diaphragms
may be applied to bring functions naturally many-valued under the application of
some of the following theorems, as Nos. 14 £f.
22
VECTOR ANALYSIS.
the expression p.p being regarded, for the present at least, as a
single operator when applied to a vector. (It will he remem¬
bered that no meaning has been attributed to p before a vec¬
tor.) It should be noticed that, if
ao=iX+jY + kZ,
fcpGo=ip.pX+jp.prY+kp.pZ,
that is, the operator p.p applied to a vector affects separately
its scalar components.
71. From the above definition with those of Flos. 52 and 54
we may easily obtain
V- V FF * 00 ~ V X v X
The effect of the operator p.p is therefore independent of
the directions of the axes used in its definition.
72. The expression — {a^p.pu, where a is any infinitesimal
scalar, evidently represents the excess of the value of the scalar
function u at the point considered above the average of its
values at six points at the following vector distances: ai,
— ai, aj, — aj, ak, — ak. Since the directions of i, j, and k are
immaterial, (provided that they are at right angles to each
other), the excess of the value of u at the central! point above
its average value in a spherical surface of radius a constructed
about that point as the center will be represented by the same
expression, — }a 2 p.pit.
Precisely the same is true of a vector function, if it is un¬
derstood that the additions and subtractions implied in the
terms average and excess are geometrical additions and sub¬
tractions.
Maxwell has called — p.pw the concentration of u, whether
u is scalar or vector. We may call p.pw (or p.po), which is
proportioned to the excess of the average value of the func¬
tion in an infinitesimal spherical surface above the value at the
center, the dispersion of u (or co).
Transformation of Definite Integrals.
73. From the equations of Ho. 65, with the principles of
integration of Hos. 57, 59, and 60, we may deduce various
transformations of definite integrals, which are entirely analo¬
gous to those known in the scalar calculus under the name of
integration by parts. The following formulae (like those of
Hos. 57, 59, and 60) are written for the case of continuous
values of the quantities (scalar and vector) to which the signs
p. p., and p X are applied. It is left to the student to complete
the formulae for cases of discontinuity in these values. The
manner in which this is to be done may in each case be inferred
VECTOR ANALYSIS.
23
from the nature of the formula itself. The most important
discontinuities of scalars are those which occur at surfaces: in
the case of vectors, discontinuities at surfaces, at lines, and at
points, should he considered.
74. From equation (3) we obtain
f 7 (tu) .dp=t"u" — t'u' —fupt.dp 4- ftpu.dp ,
where the accents distinguish the quantities relating to the
limits of the line-integrals. We are thus able to reduce a
line-integral of the form fupt.dp to the form — ftpu.dp with
quantities free from the sign of integration.
75. From equation (5) we obtain
ffp X (u go). do —Ju gj. dp —ffupX oo.da —ffGoXpu.dp ,
where, as elsewhere in these equations, the line-integral relates
to the boundary of the surface integral.
From this, by substitution of pt for to, we may derive as a
particular case
ffpux pt.dff=fupt.dp= —ftpu.dp.
76. From equation (4) we obtain
Iff (7- \ugd\ dv =/Juoo.da =fff go. pu dv + fffup. go dv,
where, as elsewhere in these equations, the surface-integral
relates to the boundary of the volume-integrals.
From this, by substitution of pt for to, we derive as a partic¬
ular case
fffpt.pu dv^ffupt.do-fffup. V tdv =jftpu.dG-ffftp.pu dv,
which is Green’s Theorem. The substitution of spt for to
gives the more general form of this theorem which is due to
Thomson, viz:—
fffspt.pu dv —ffusp t.dtr —fffu p.[spt]dv
—ff^pu.da—ffftp. [spu]dv.
77. From equation (6) we obtain
Jlfv\ r X oo]dv=ffr X Go.dG=fffoo.pX r dv-fffr.p X go dv.
A particular case is
P X oo dv —ffoo x pu.dtJ.
Integration of Differential Equations.
78. If throughout any continuous space (or in all space)
pu—0,
24
VECTOR ANALYSIS.
then throughout the same space
u— constant.
79. If throughout any continuous space (or in all space)
/ 7 . /7 'V> —■ 0 ,
and in any finite part of that space, or in any finite surface in
or bounding it,
pu= 0,
then throughout the whole space
yu— 0, and u— constant.
This will appear from the following considerations.
If yn~ 0 in any finite part of the space, u is constant in that
part. If u is not constant throughout, let us imagine a sphere
situated principally in the part in which u is constant, hut pro¬
jecting slightly into a part in which u has a greater value, or
else into a part in which u has a less. The surface-integral of
yu for the part of the spherical surface in the region where
u is constant will have the value zero : for the other part of
the surface, the integral will be either greater than zero, or less
than zero. Therefore the whole surface-integral for the spher¬
ical surface will not have the value zero, which is required by
the general condition, p.p’-u— 0.
Again, if yu =0 only in a surface in or bounding the space
in which y.yu — 0, u will be constant in this surface, and the
surface will be contiguous to a region in which y.yu —0 and u
has a greater value than in the surface, or else a less value
than in the surface. Let us imagine a sphere lying principally
on the other side of the surface, but projecting slightly into
this region, and let us particularly consider the surface-integral
of yu for the small segment cut off by the surface yu— 0. The
integral for that part of the surface of the segment which con¬
sists of part of the surface yu —0 will have the value zero, the
integral for the spherical part will have a value either greater
than zero or else less than zero. Therefore the integral for the
whole surface of the segment cannot have the value zero,
which is demanded by the general condition, r-ru=0.
80. If throughout' a certain space (which need not be con¬
tinuous, and which may extend to infinity)
y.yu— 0,
and in all the bounding surfaces
u— constant=«,
and (in case the space extends to infinity) if at infinite dist-
VECTOR ANALYSIS.
25
ances within the space u=a ,—then throughout the space
pu=0, and u—ci.
For, if anywhere in the interior of the space pu has a value
ditferent from zero, we may find a point P where such is the
case, and where u has a value b different from a, —to fix our
ideas we will say less. Imagine a surface enclosing all of the
space in which u < b. (This must he possible, since that part of
the space does not reach to infinity.) The surface-integral of
pu for this surface has the value zero in virtue of the general
condition p.pic= 0. Put, from the manner in which the surface
is defined, no part of the integral can be negative. Therefore
no part of the integral can be positive, and the supposition
made with respect to the point P is untenable. That the sup¬
position that b > a is untenable may be shown in a similar man¬
ner. Therefore the value of u is constant.
This proposition may be generalized by substituting the con¬
dition p.[tpu] = 0 for p.pu=z 0, t denoting any positive (or any
negative) scalar function of position in space. The conclusion
would be the same, and the demonstration similar.
81. If throughout a certain space (which need not be con¬
tinuous, and which may extend to infinity,)
p.pu— 0,
and in all the bounding surfaces the normal component of pu
vanishes, and at infinite distances within the space (if such
there are) r 2 — = 0, where r denotes the distance from a fixed
dr
origin, then throughout the space
pu — 0,
and in each continuous portion of the same
u= constant.
For, if anywhere in the space in question pu has a value
different from zero, let it have such a value at a point P, and
let u be there equal to b. Imagine a spherical surface about
the above-mentioned origin as center, enclosing the point P,
and with a radius r. Consider that portion of the space to
which the theorem relates which is within the sphere and in
which u<db. The surface-integral of pu for this space is equal
to zero in virtue of the general condition p.pu— 0. That part
of the integral (if any) which relates to a portion of the
spherical surface has a value numerically not greater than
4 nr‘
where
denotes the greatest numerical value
du
of — in the portion of the spherical surface considered.
26
VECTOR ANALYSIS.
Hence, the value of this part of the surface-integral may be
made less (numerically) than any assignable quantity by giving
to r a sufficiently great value. Hence, the other part of the
surface-integral (viz., that relating to the surface in which
u—h, and to the boundary of the space to which the theorem
relates,) may be given a value differing from zero by less than
any assignable quantity. But no part of the integral relating
to this surface can be negative. Therefore no part can be
positive, and the supposition relative to the point P is unten¬
able.
This proposition also may be generalized by substituting
C^'l f $AJj
[7.\tf7u] = 0 for f7.pu= 0, and tr 2 — = 0 for r 3 —=0.
Chi* - (ajV
82. If throughout any continuous space (or in all space)
\71=17U ,
then throughout the same space
t—u-\- const.
The truth of this and the three following theorems will be
apparent if we consider the difference t—u.
83. If throughout any continuous space (or in all space)
f7. pt=p.pu,
and in any finite part of that space, or in any finite surface in
or bounding it,
Vt=pu ,
then throughout the whole space
and t=u + const.
84. If throughout a certain space (which need not be con¬
tinuous, and which may extend to infinity)
/7.p2=£7./7W,
and in all the bounding surfaces
t—u ,
and at infinite distances within the space (if such there are)
t=u ,
then throughout the space
t — U.
85. If throughout a certain space (which need not be con¬
tinuous, and which may extend to infinity)
fi',
VECTOR ANALYSIS.
27
and in all the bounding surfaces the normal components of
ft and fu are equal, and at infinite distances within the space
(if such there are) r2 (~^ r ~~ ^^=0, where r denotes the distance
from some fixed origin,—then throughout the space
ft—fU,
and in each continuous part of which the space consists
fc*u = constant.
86. If throughout any continuous space (or in all space)
fXr—fXoo and f.r = j/.Go,
and in any finite part of that space, or in any finite surface in
or bounding it,
T=GO,
then throughout the whole space
T— 00.
For, since fX(z—co)=0, we may set fu=z—co, making the
space acyclic (if necessary) by diaphragms. Then in the whole
space u is single-valued and f.fu— 0, and in a part of the space,
or in a surface in or bounding it, fu— 0. Flence throughout
the space fu—z—co—0.
87. If throughout an aperiphractic* space contained within
finite boundaries but not necessarily continuous
(7Xi=(7X® and f.r = f.co,
and in all the bounding surfaces the tangential components of
z and co are equal, then throughout the space
T — GO.
It is evidently sufficient to prove this proposition for a con¬
tinuous space. Setting fu=z—co , we have f.fu= 0 for the
whole space, and u —constant for its boundary, which will be a
single surface for a continuous aperiphractic space. ITence
throughout the space fu=z—co=0 .
88. If throughout an acyclic space contained within finite
boundaries but not necessarily continuous
fX z—fX go and f.r = f.GD ,
and in all the bounding surfaces the normal components of r
and co are equal, then throughout the whole space
r = go.
* If a. space encloses within itself another space, it is called periphradic , other¬
wise aperiphraclic.
28
YECTOE ANALYSIS.
Setting f/'U—T—o), we have 0 throughout tlie space,
and tlie normal component of pu at tire boundary equal to
zero. Hence throughout the whole space pu—T—w= 0.
89. If throughout a certain space (which need not be con¬
tinuous, and which may extend to infinity)
p.\7T — \7.\7GQ
and in all the bounding surfaces
7—GO ,
and at infinite distances within the space (if such there are)
T=GO,
then throughout the whole space
r— go.
This will be apparent if we consider separately each of the
scalar components of r and co.
Minimum Yalues of the Volume-integral fjfu co.co dr.
{Thomson' 1 s Theorems .)
90. Let it be required to determine for a certain space a
vector function of position co subject to certain conditions (to
be specified hereafter), so that the volume-integral
fffu go. go dv
for that space shall have a minimum value, u denoting a given
positive scalar function of position.
a. In the first place, let the vector co be subject to the con¬
ditions that p.co is given within the space, and that the nor¬
mal component of co is given for the bounding surface. (This
component must of course be such that the surface-integral of
co shall be equal to the volume-integral fp.codv. If the space
is not continuous, this must be true of each continuous portion
of it. See No. 57.) The solution is that px(uco)= 0, or more
generally, that the line-integral of uco for any closed curve in
the space shall vanish.
The existence of the minimum requires that
fff u go. 8go dv — 0,
while do is subject to the limitation that
pr.dGO=0,
and that the normal component of oco at the bounding surface
vanishes. To prove that the line-integral of uco vanishes for
VECTOR ANALYSIS.
29
any closed curve within the space, let ns imagine the curve to
be surrounded by an infinitely slender tube of normal section
dz, which may be either constant or variable. We may satisfy
the equation p,dco~ 0 by making dco =0 outside of the tube,
and dcodz=da^~ within it, da denoting an arbitrary infinitesimal
constant, p the position-vector, and ds an element of the length
of the tube or closed curve. We have then
fffu go. 6 go dv—fu od.6 go dz ds—fu oo.dp 6a—6a fit Go.dp=0 ,
whence fu oo.dp= 0 . q. e. d.
We may express this result by saying that uco is the derivative
of a single-valued scalar function of position in space. (See
ISTo. 67.)
If for certain parts of the surface the normal component of
co is not given for each point, but only the surface-integral of
co for each such part, then the above reasoning will apply not
only to closed curves, but also to curves commencing and end¬
ing in such a part of the surface. The primitive of uco will
then have a constant value in each such part.
If the space extends to infinity and there is no special condi¬
tion respecting the value of co at infinite distances, the prim¬
itive of uco will have a constant value at infinite distances
within the space or within each separate continuous part of it.
If we except those cases in which the problem has no defin¬
ite meaning because the data are such that the integral
fuco.codv must be infinite, it is evident that a minimum must
always exist, and (on account of the quadratic form of the
integral) that it is unique. That the conditions just found are
sufficient to insure this minimum, is evident from the consider¬
ation that any allowable values of dco may be made up of such
values as we have supposed. Therefore, there will be one and
only one vector function of position in space which satisfies
these conditions together with those enumerated at the begin¬
ning of this number.
b. In the second place, let the vector co be subject to the
conditions that /7 X co is given throughout the space, and that
the tangential component of co is given at the bounding sur¬
face. The solution is that
17.[u go]=:0,
and, if the space is periphractic, that the surface-integral of uco
vanishes for each of the bounding surfaces.
The existence of the minimum requires that
fffu 00.600 dv = 0,
%
30
VECTOR ANALYSIS.
while ooj is subject to the conditions that
(7X<5go= 0 ,
and that the tangential component of dco in the hounding sur¬
face vanishes. In virtue of these conditions we may set
Sgd— f/Sq,
where dq is an arbitrary infinitesimal scalar function of posi¬
tion, subject only to the condition that it is constant in each of
the bounding surfaces. (See Ho. 67.) By substitution of this
value we obtain
fffu GD.pSq dv=0,
or integrating by parts (Ho. 76)
ff u gd.cJij dq—JJ]‘{7.[u Go]dq dv =0.
Since dq is arbitrary in the volume-integral, we have through¬
out the whole space
( 7 .[u <»] = 0;
and since dq has an arbitrary constant value in each of the
bounding surfaces (if the boundary of the space consists of
separate parts), we have for each such part
ffu Go.dc> — 0 .
Potentials , Newtonians , Laplacians.
91. Pef .—If u' is the scalar quantity of something situated
at a certain point p f , the potential of u' for any point p is a
scalar function of p, defined by the equation
pot? A
u
[p'-p]<
and the Hewtonian of u' for any point p is a vector function of
p defined by the equation
new u'—Y^i —Ws u '-
[p — P]o
Again, if ay is the vector representing the quantity and
direction of something situated at the point //, the potential
and the Laplacian of at' for any point p are vector functions of
p defined by the equations
poW= [^>y
VECTOR ANALYSIS.
31
92. If u or (o is a scalar or vector function of position in
space, we may write Potw, KewM, Potw, Lapo for the vol¬
ume-integrals of pot u', etc., taken as functions of <>'; i. e. we
may set
Pot u=fff pot y! dv 1 =fff~—^dv>\
New u=ff/Tie w u’ dv'v! dv',
Pot oo^fff pot go' dv’=Jff^j~^dv',
Lap ai=fff lap go' dv'=f/f ^^- x go' dv ',
where the p is to he regarded as constant in the integration.
This extends over all space, or wherever the u' or co' have any
values other than zero. These integrals may themselves be
called (integral) potentials, Newtonians, and Laplaeians.
93.
<^PotM_ T) jdu
~7hT~ 0t (kd
d P ot go doo
dx ~ °^dx ‘
This will he evident with respect both to scalar and to vector
functions, if we suppose that when we differentiate the poten¬
tial with respect to x, (thus varying the position of the point
for which the potential is taken) each element of volume dv' in
the implied integral remains fixed, not in absolute position ,
but in position relative to the point for which the potential is
taken. This supposition is evidently allowable whenever the
integration indicated by the symbol Pot tends to a definite
limit when the limits of integration are indefinitely extended.
Since we may substitute y and & for x in the preceding
formula, and since a constant factor of any kind may be intro¬
duced under the sign of integration, we have
p Pot w=Pot pu
p.Pot G3 — Pot p.co
/7 X Pot 03 —Pot pXGO
p>p Pot u=Yot p.p$ LA^
p.p Pot 03 — Pot p.pGO
i. e., the symbols p, p., pX, p.p may be applied indifferently
before or after the sign Pot.
Yet a certain restriction is to be observed. When the oper¬
ation of taking the (integral) potential does not give a definite
finite value, the first members of these equations are to be
regarded as entirely indeterminate, but the second members
may have perfectly definite values. This would be the case
for example, if u or co had a constant value throughout all
32
VECTOR ANALYSIS.
space. It might seem harmless to set an indefinite expression
equal to a definite, hut it would he dangerous, since we might
with equal right set the indefinite expression equal to other
definite expressions, and then be misled into supposing these
definite expressions to be equal to one another. It will be safe
to say that the above equations will hold, provided that the
potential of u or to has a definite value. It will be observed
that whenever Pot u or Pot to lias a definite value in general ,
(i. e. with the possible exception of certain points, lines, and
surfaces),* the first members of all these equations will have
definite values in general, and therefore the second members
of the equations, being necessarily equal to the first members,
when these have definite values, will also have definite values
in general.
94. Again, whenever Pot u has a definite value, we may
write
u!
V Pot u =Vfff-pdv'=ffJ'p- u' dv'.
where r stands for [//—/>]„. But
Avbence
1 P'-P
7 ~——w
y g • iyo
17 Pot WzrrNeW U.
Moreover, ISTew u will in general have a definite value, if
Pot u has.
95. In like manner, whenever Pot to has a definite value,
7 X «*=V X ~ dv’=fff 7 y X of dv'.
Substituting the value of 7 —
given above we have
7 X Pot £e|f=Lap GO.
Lap to will have a definite value in general, whenever Pot to
has.
96. Hence, with the aid of No. 93, we obtain
7 X Lap go= Lap pX go,
7 . Lap go= 0.
whenever Pot to has a definite value.
97. By the method of No. 93 we obtain
7 . N ewu=7. dv l —fff [7 u r .
* Whenever it is said that a function of position in space has a definite value
in general , this phrase is to he understood as explained above. The term definite
is intended to exclude both indeterminate and infinite values.
VECTOR ANALYSIS.
33
To find the value of this integral, we may regard the point p,
which is constant in the integration, as the center of polar
coordinates. Then r becomes the radius vector of the point p',
and we may set
dv'=r 2 clq dr,
where r-dq is the element of a spherical surface having center
at p and radius r. We may also set
We thus obtain
p' — p_du'
r ~~ dr'
p.New u=py~ dq dr=Anf-~dr=Ami' r==00
4 7tW r=() ,
where u denotes the average value of u in a spherical surface
of radius r about the point y i( as center.
Now if Pot u has in general a definite value, we must have
u' — 0 for r= co . Also, p.New u will have in general a defin¬
ite value. For r— 0 , the value of u' is evidently u. We have,
therefore,
£7.New u——Anu,
£7. £7 Pot U- — Ann.
98. If Pot co has in general a definite value,
£7. £7 Pot 09= £7. £7|> i + Vj+W k] = p.yu % + £L pvj + £7. £7«> k,
£ 7.£7 Pot 09=— Anoo.
ITence, by No. 71,
£7 X £7 X Pot GO — £7 £7.Pot GO — 4 7T GO.
That is,
Lap £7Xo5— New [/.Go—Anco.
If we set
1 T — 1 AT
go =^~ Lap i/X go, f» 2 =— New £ 7 . 03 ,
we have
03= G0 X + 03 2 ,
where co 1 and co. 2 are such functions of position that [ 7 .( 0 , = 0, and
£7 Xc3 2 =0. This is expressed by saying that co l is solenoidcil ,
and op irrotational. Pot op and Pot op, like Pot co, will have
in general definite values.
It is worth while to notice that there is only one way in
which a vector function of position in space having a definite
potential can be thus divided into solenoidal and irrotational
parts having definite potentials. For if o), + e, co 2 —e are two
other such parts,
\7 • 0 and £7 x £=0,
5
34
VECTOR ANALYSIS.
Moreover, Pot e lias in general a definite value, and therefore
1 T 1
£=—Lap J7[|< .& — - — .New [ 7 . 8 = 0 . q. e. d.
4 7t 4 71 v
99. To assist the memory of the student, some of the princi¬
pal results of Nos. 93-98 may be expressed as follows :
Let w 1 he any solenoidal vector function of position in space,
o, any irrotational vector function, and u any scalar function,
satisfying the conditions that their potentials have in general
definite values.
With respect to the solenoidal function cy, — Lap and pX
are inverse operators ; i. e.,
1 T 1 T
— Lap ! 7 X 00=7 X— Lap
Applied to the irrotational function <v 2 , either of these opera¬
tors gives zero ; i. e.,
Lap a) 2 =0, pX© 2 =0.
With respect to the irrotational function ca 2 , or the scalar func¬
tion u, -U New and — p. are inverse operators ; i. e.,
4 7t
i X
—— New [7.od= g% — 77 .— New u—u.
'4 n 47T
Applied to the solenoidal function co 1 the operator.p. gives
zero; i. e.,
[ 7 . 00 = 0 ,
Since the most general form of a vector function having in
general a definite potential may be written co, + a)„ the effect of
these operators on such a function needs no especial mention.
With respect to the solenoidal function a> l9 — Pot and
pXpX are inverse operators; i. e.,
Pot (7X{7 X®!=/7X^ Pot j 7 X oo t —v X V X ~ Pot go 1 = co v
With respect to the irrotational function cj 2 , ~ Pot and
—pp. are inverse operators; i. e.,
Pot p y.oo g ^ -F^Potfziyg = - p p.— Pot a?* = .
With respect to any scalar or vector function having in gen¬
eral a definite potential — Pot and -p.p are inverse opera¬
tors ; i. e.,
VECTOR ANALYSIS.
35
1
47T
Pot [ 7 .p?u= — [7. J- Pot pu= — p.p—Pot U—U,
1
4 71
Pot p.p [dJ 1 + GJ 2 ]=—p.p
1
4 7T
Pot [uq
+ — GJj + Ce7g.
With respect to the solenoidal function w,, ~p ./7 and pXpX
are equivalent: with respect to the irrotational function cq
p.p and p p. are equivalent; i. e.,
j7.p 2 = 7p(y 2
100 . the interpretation of the preceding formulae .—
Infinite values of the quantity which occurs in a volume-inte¬
gral as the coefficient of the element of volume will not neces¬
sarily make the value of the integral infinite, when they are
confined to certain surfaces, lines, or points. Yet these sur¬
faces, lines, or points may contribute a certain finite amount
to the value of the volume-integral, which must he separately
calculated, and in the case of surfaces or lines is naturally
expressed as a surface- or line-integral. Such cases are easily
treated by substituting for the surface, line, or point, a very
thin shell, or filament, or a solid very small in all dimensions,
within which the function may be supposed to have a very
large value.
The only cases which we shall here consider in detail are
those of surfaces at which the functions of position (u or co)
are discontinuous, and the values of pu, p x w, p.co thus
become infinite. Let the function u have the value u 1 on the
side of the surface which we regard as the negative, and
the value u 2 on the positive side. Let Au—u 2 — u 1 . If we
substitute for the surface a shell of very small thickness a,
within which the value of u varies uniformly as we pass
through the shell, we shall have pu=v — within the shell
a ’
v denoting a unit normal on the positive side of the surface.
The elements of volume which compose the shell may be ex¬
pressed by a[daf where [>/u ] 0 is the magnitude of an element
of the surface, clo being the vector element. Hence,
pu dv = r Au \df 0 — Au da.
Hence, when there are surfaces at which the values of u are
discontinuous, the full value of Pot pu should always be under¬
stood as including the surface-integral
/V ^ U ' 71 '
f/ UmprP
relating to such surfaces. (Aid and da' are accented in the
formula to indicate that they relate to the point f.)
36
VECTOR ANALYSIS.
In the case of a vector function which is discontinuous at a
surface, the expressions p.codv and pXtodv, relating to the
element of the shell which we substitute for the surface of dis¬
continuity, are easily transformed by the principle that these
expressions are the direct and skew surface-integrals of to for
the element of the shell. (See Hos. 55, 56.) The part of the
surface-integrals relating to the edge of the element may evi¬
dently he neglected, and we shall have
j7 . go dv — co^.da— oo i .da= A co.da,
j 7 X go dv=dax co^—dax Go^—dffxAoo.
Whenever, therefore, to is discontinuous at surfaces, the
expressions Pot p.to and Hew f .to must be regarded as implic¬
itly including the surface-integrals
//r
[p'-p]
-A go'. da'
and
AA f
p -p
[p-p]
3 Ago' .da 1
0
respectively, relating to such surfaces, and the expressions
Pot [?Xto and Lap pXto as including the surface-integrals
and JT \x^k x[da ' xAw ' ]
respectively, relating to such surfaces.
101. We have already seen that if to is the curl of any vec¬
tor function of position, p.to= 0. (Ho: 68.) The converse is
evidently true, whenever the equation \ 7 .to —0 holds through¬
out all space, and to has in general a definite potential; for then
go=!7 X — Lap go.
K 47T 1
Again, if [7.to = 0 within any aperiphractic space A, contained
within finite boundaries, we may suppose that space to be en¬
closed by a shell B having its inner surface coincident with the
surface of A. We may imagine a function of position to', such
that to' — to in A, to' = 0 outside of the shell B, and the integral
fffto'.oo'dv for B has the least value consistent with the con¬
ditions that the normal component of to' at the outer surface is
zero, and at the inner surface is equal to that of to. Then
pr,co'—0 throughout all space, (Ho. 90,) and the potential of to'
will have in general a definite value. Lienee,
m = / 7 X^; Lap go ,
and to will have the same value within the space A.
New Haven: Printed by Tuttle, Morehouse & Taylor, 1881.
VECTOR ANALYSIS.
37
102. Def .—If co is a vector function of position in space,
the Maxwellian * of co is a scalar function of position defined
by the equation
Max go — rrr/-, — • &)! ^ v '-
' [p'-p ] 3 0
(Compare No. 92.) From this definition the following prop¬
erties are easily derived. It is supposed that the functions co
and u are such that their potentials have in general definite
values.
Max go — pr-. Pot go — Pot /7. go ,
p Max go — 1717 . Pot go — New [ 7 . go,
Max [ 7 ti — —4 nu,
An go = [7 x Fap go — [7 Max go.
If the values of Lap Lap co, New Max co, and Max New u are
in general definite, we may add
47r Pot go — Lap Lap go — New Max go,
4 7 t Pot u — — Max New u.
In other words:—-The Maxwellian is the divergence of the
potential, —and [7 are inverse operators for scalars and
irrotatioual vectors, for vectors in general —— u Max is an
An 1
operator which separates the irrotatioual from the solenoidal
part. For scalars and irrotatioual vectors, 1 Max New and
An
^ J New Max give the potential, for solenoidal vectors Lap
Lap gives the potential, for vectors in general — New Max
47T
gives the potential of the irrotatioual part, and ~~ Lap Lap the
potential of the solenoidal part.
103. Def. —The following double volume-integrals are of
frequent occurrence in physical problems. They are all scalar
quantities, and none of them functions of position in space, as
are the single volume-integrals which we have been consid¬
ering. The integrations extend over all space, or as far as the
expression to he integrated has values other than zero.
* The frequent occurrence of the integral in Maxwell’s Treatise on Electricity
and Magnetism has suggested this name.
6
38
VICTOR ANALYSIS.
The mutual potential , or potential product , of two scalar
functions of position in space, is defined by the equation
u w 1
Pot (u, w) — ffffff —ydv dv' = fffu Pot w dv = fff w Pot u dv.
In the double volume-integral, r is the distance between the
two elements of volume, and u relates to dv as w' to dv'.
The mutual potential , or potential product , of two vector
functions of position in space, is defined by the equation
Pot (q>, CO) dv dv'
— fff pNot go dv — fff oo .Pot cp dv.
The mutual Laplacian , or Laplacian product , of two
vector functions of position in space, is defined by the
equation
Lap (cp, go) = ffffff go dv dv'
—fff 00 - P a P P dv ~ f/fP-^V 00 dv.
The Newtonian product of a scalar and a vector function of
position in space is defined by the equation
New (u , go) = ffffff oo f 3 - u' dv dv’—fff ttf.New u dv.
The Maxwellian product of a vector and a scalar function
of position in space is defined by the equation
Max (go, u) = ffffff u 9 J- -. go ' dv dv'
— fff u M ax go dv = — New (u, go).
It is of course supposed that u, w, cp, to are such functions of
position that the above expressions have definite values.
104. By No. 97,
4:7ZU Pot W r= — £ 7 .New u Pot w
— — p,[New u Pot ui\ -j- New u. New w.
■#
The volume-integral of this equation gives
47 T Pot (u, w) = fff New u. New w dv,
if the integral
ffdff. N ew u Pot w
VECTOR ANALYSIS. 39
for a closed surface, vanislies when the space included by the
surface is indefinitely extended in all directions. This will he
the case when everywhere outside of certain assignable limits
the values of u and w are zero.
Again, by No. 102,
47zr®.Pot cp = /7 xLap co • Pot (p —[7 Max ®.Pot cp
— 17. [Lap go X Pot cp\ + Lap go . Lap cp
— / 7 .[Max go Pot <p] + Max go Max cp.
The volume-integral of this equation gives
47zr Pot (cp, go) — fff Lap cp . Lap® du + fff Max cp Max go dv,
if the integrals
ff dff . Lap go X Pot cp, ff do '. Pot cp Max go,
for a closed surface, vanish when the space included by the
surface is indefinitely extended in all directions. This will be
the case if everywhere outside of certain assignable limits the
values of cp and go are zero.
40
VECTOR ANALYSIS.
CHAPTER III.
CONCERNING LINEAR VECTOR FUNCTIONS.
105. Def .—A vector function of a vector is said to be
linear , when the function of the sum of any two vectors is
equal to the sum of the functions of the vectors. That is, if
func.[p + p'] — func.[p] +func.[p']
for all values of p and //, the function is linear. In such cases
it is easily shown that
func. [ap + bp' -f op" + etc.]
= a func. \p\ + b func.[p'] + c func. [p"] + etc.
106. An expression of the form
a A.p + yd p.p-f etc.
evidently represents a linear function of p, and may he con¬
veniently written in the form
The expression
or
|«A + ySp 4-etc. }.p.
p.aX + p./3 p-f-etc.,
p. {o'A + ftp + etc.},
also represents a linear function of p, which is, in general,
different from the preceding, and will be called its conjugate .
107. Def .—An expression of the form al or ftp will be
called a dyad. An expression consisting of any number of
dyads united by the signs + or — will be called a dyadic bino¬
mial, trinomial , etc., as the case may be, or more briefly, a
dyadic. The latter term will be used so as to include the case
of a single dyad. When we desire to express a dyadic by a
single letter, the Greek capitals will be used, except such as
are like the Roman, and also A and 1\ The letter I will also
be used to represent a certain dyadic, to be mentioned hereafter.
Since any linear vector function may be expressed by means
of a dyadic, (as we shall see more particularly hereafter, see
Ho. 110,) the study of such functions, which is evidently of
primary importance in the theory of vectors, may be reduced
to that of dyadics.
VECTOR ANALYSIS.
41
108. Def .—Any two dyadics 0 and W are equal,
when 0 .p = W.p for all values of p,
or, when p. 0 = p. W for all values of p,
or, when 0.0.p — 0. W.p for all values of 0 and of p.
The third condition is easily shown to he equivalent both to
the first and to the second. The three conditions are therefore
equivalent.
It follows that 0=¥,if 0.p= W.p, or p,0—p.W , for three
non-complanar values of p.
109. Def .—We shall call the vector 0.p the (direct) product
of 0 and p, the vector p. 0 the (direct) product of p and 0, and
the scalar a. 0.p the (direct) product of a, 0 , and p.
In the combination 0.p , we shall say that 0 is used as a
prefactor, in the combination p. 0, as a postfactor.
110. If r is any linear function of p, and for p — i, p=j, p~l\
the values of r are respectively a, /9, and y, we may set
and also
r ={ai + fij+yJc}.p, .
r — p.{ ia +j/3 + ky \. ~T—
- 2 - ^ ~ ~ ~
I Rf >
Therefore, any linear function may be expressed by a dyadic
as prefactor and also by a dyadic as postfactor.
111. Def .—We shall say that a dyadic is multiplied by a
scalar, when one of the vectors of each of its component dyads
is multiplied by that scalar. It is evidently immaterial to
which vector of any dyad the scalar factor is applied. The
product of the dyadic 0 and the scalar a may be written either
a0 or 0 a. The minus sign before a dyadic reverses the signs
of all its terms.
112. The sign + in a dyadic, or connecting dyadics, may be
regarded as expressing addition, since the combination of
dyads and dyadics with this sign is subject to the laws of asso¬
ciation and commutation.
113. The combination of vectors in a dyad is evidently dis¬
tributive. That is,
-
-f K-f>ks
P r=. V+C.C
V + W-J
+ V+k.k.
— —
[o'-!- (3 + etc.] [A -|-p + etc.] = pA + + /9A-f fip +etc.
We may therefore regard the dyad as a kind of product of the
two vectors of which it is formed. Since this kind of product
is not commutative, we shall have occasion to distinguish the
factors as antecedent and consequent.
114. Since any vector may be expressed as a sum of i,j, and
h with scalar coefficients, every dyadic may be reduced to a
sum of the nine dyads
n, ij, ik, ji, J, jk, hi, kp kk.
12
VECTOR ANALYSIS.
(f
a
Qr^yUti'J
4
*•
with scalar coefficients. Two such sums cannot be equal
according to the definitions of No. 108, unless their coefficients
are equal each to each. Hence dyadics are equal only when
their equality can he deduced from the principle that the
operation of forming a dyad is a distributive one.
On this account, we may regard the dyad as the most gen¬
eral formmf product of two vectors. We shall call it the'Tnde-
terminate product. The complete determination of a single
dyad involves six independent scalars, of a dyadic, nine.
115. It follows from the principles of the last paragraph
that if
2a fi = 2kA,
then
2axfi = 2 xX A,
and
2a.fi = 2 k. A.
In other words, the vector and the scalar obtained from a
dyadic by insertion of the sign of skew or direct multiplication
in each dyad are both independent of the particular form in
which the dyadic is expressed.
We shall write 0 x and 0 B to indicate the vector and the
scalar thus obtained.
\
0 x — (]. 0.7c—7c. 0. j) i -f- (7c. 0. i — i. 0.7c) j + ( i. 0,j —j. 0. i ) 7c,
0 S = i. 0. i + j. 0. j + 7c, 0.7c,
as is at once evident, if we suppose 0 to be expanded in terms
of ii, ij , etc.
116. Def. —The {direct) product of two dyads (indicated by
a dot) is the dyad formed of the first and last of the four fac¬
tors, multiplied by the direct product of the second and third.
That is,
\afi\.\ yd j- — a fi.y d — fi.y ad.
The (direct) product of two dyadics is the sum of all the jiro-
ducts formed by prefixing a term of the first dyadic to a term
of the second. Since the direct product of one dyadic with
another is a dyadic, it may be multiplied in the same way by a
third, and so on indefinitely. This kind of multiplication is
evidently associative, as well as distributive. The same is true
of the direct product of a series of factors of which the first
and the last are either dyadics or vectors, and the other factors
are dyadics. Thus the values of the expressions
a.0.G.W./3 , a.0.G, 0.0.W.fi , 0 .G.W
will not be affected by any insertion of parentheses. But this
VECTOR ANALYSIS.
43
kind of multiplication is not commutative, except in the case
of the direct product of two vectors.
117. Def .—The expressions 0 Xp and pX 0 represent dyad-
ics which we shall call the skew products of 0 and p. If
0 = a\ + ft p + e tc.,
these skew products are defined by the equations
0 X p — ol A X p 4- £> p X p + etc. ,
p x 0 = p X ol A + p X ft p + etc.
*
It is evident that
) p x 0 }. W = p X { 0 • W 1 , V.{§ xp} = \¥.0\xp,
\pX 0}-a — px\_0.ot\, a.{ 0xp} = [a.0]xp,
IPX 0} X&= PX { t 0Xa\.
We may therefore write without ambiguity
p X r ^. W, W. <*> X p, p X or, m ® X p, p X o'.
(VEc) e ~Ve (oe)
(YEc.) f - Ve(lF)
V{yec)F- vtVcF
This may be expressed a little more generally by saying that
the associative principle enunciated in Ho. 116 may be ex¬
tended to cases in which the initial or final vectors are con¬
nected with the other factors by the sign of skew multiplication.
Moreover,
a . p x = [OL X p\ ■ <jtJ au d { I J X p- <x = 0. [pX o'].
These expressions evidently represent vectors. Bo
ripx wxp\. ( i >.
FVEc =
(VcE)Ft* c.(VEf)
These expressions represent dyadics. The braces cannot be
omitted without ambiguity.
118. Since all the antecedents or all the consequents in any
dyadic may be expressed in parts of any three non-complanar
vectors, and since the sum of any number of dyads having the
same antecedent or the same consequent may be expressed by
a single dyad, it follows that any dyadic may be expressed as
the sum of three dyads, and so, that either the antecedents or
the consequents shall be any desired non-complanar vectors,
but only in one wav when either the antecedents or the conse-
v 9J
quents are thus given.
In particular, the dyadic *
aii + bij + oik
T o ji + bjj + c'jk
+ a"ki+b"kj + o"kk,
44
VECTOR ANALYSIS.
which may for brevity he written
and to
where
is equal to
ai + {3j +yk,
where
a — ai + a j + a"k,
(3 = hi + b'j + b"k,
y — ci+c'j+c"k,
iX +jp + kv,
A = a i + bj + ck
pi = a' i + b''j + c’k ,
v = a"i + b"J+c"k.
119. By a similar process, the sum of three dyads may he
reduced to the sum of two dyads, whenever either the antece¬
dents or the consequents are complanar, and only in such
cases. To prove the latter point, let us suppose that in the
dyadic
ocX -f m y + y v
neither the antecedents nor the consequents are complanar.
The vector
{o'A-f ji )i + y r |. p
is a linear function of p which will he paralle] to a when p is
perpendicular to p and v, which will he parallel to ft when p is
perpendicular to v and A, and which will he parallel to y when
p is perpendicular to A and a. Hence, the function may he
given any value whatever by giving the proper value to p.
This would evidently not he the case with tlie sum of two
dyads. Hence, by Ho. 108, this dyadic cannot he equal to the
sum of two dyads.
120. In like manner, the sum of two dyads may he reduced
to a single dyad, if either the antecedents or the consequents are
parallel, and only in such cases.
A sum of three dyads cannot he reduced to a single dyad,
unless either their antecedents or consequents are parallel, or
both antecedents and consequents are (separately) complanar.
In the first case the reduction can always he made, in the second,
occasionally.
121. Def. —A dyadic which cannot he reduced to the sum
of less than three dyads will he called complete.
VECTOR ANALYSIS,
45
A dyadic which can be reduced to the sum of two dyads
will be called planar. When the plane of the antecedents
coincides with that of the consequents, the dyadic will be
called unbplanar. These planes are invariable for a given
dyadic, although the dyadic may be so expressed that either
the two antecedents or the two consequents may have any
desired values (which are not parallel) within their planes.
A dyadic which can be reduced to a single dyad will be
called linear. When the antecedent and consequent are paral¬
lel, it will be called unilinear.
A dyadic is said to have the value zero, when all its terms
vanish.
122. If we set
Ur=:#.p ? T = p.ft,
and give p all possible values, a and r will receive all possible
values, if <l> is complete. The values of a and r will be con¬
fined each to a plane, if 0 is planar, which planes will coincide,
if 0 is uniplanar. The values of a and r will be confined each
to a line if 0 is linear, which lines will coincide, if 0 is uni¬
linear.
123. The products of complete dyadics are complete, of
complete and planar dyadics are planar, of complete and linear
dyadics are linear.
The products of planar dyadics are planar, except that when
the plane of the consequents of the first dyadic is perpendicular
to the plane of the antecedents of the second dyadic, the prod¬
uct reduces to a linear dyadic.
The products of linear dyadics are linear, except that when
the consequent of the first is perpendicular to the antecedent
of the second, the product reduces to zero.
The products of planar and linear dyadics are linear, except
when, the planar preceding, the plane of its consequents is per¬
pendicular to the antecedent of the linear, or, the linear pre¬
ceding, its consequent is perpendicular to the plane of the
antecedents of the planar. In these cases the product is zero.
All these cases are readily proved, if we set
o'— ft.v.p,
and consider the limits within which a varies, when we give p
all possible values.
The products Wxp and pX 0 are evidently planar dyadics.
124. Def .—A dyadic 0 is said to be an idemfactor, when
ft.p— p for all values of p,
or when p. 0> — p for all values of p.
46
VECTOR AXALYSIS,
If either of these conditions holds true, 0 must he reducible to
the form
ii -\~jj + f'k.
Therefore, both conditions will hold, if either do. All such
dyadics are equal, by ho. 108. They will be represented by
the letter I.
The direct product of an idem factor with another dyadic is
equal to that dyadic. That is,
I. <b — <I J , ']>. I r=
where 0 is any dyadic.
A dyadic of the form
aa' + (ip 1 +
in which a', [i\ y' are the reciprocals of a, p, y, is an idemfactor.
(See No. 38.) A dyadic trinomial cannot be an idemfactor, un¬
less its antecedents and consequents are reciprocals.
125. If one of the direct products of two dyadics is an idem-
factor, the other is also. For, if 0. W=I,
(7. <i>. W = (7
for all values of <r, and 0 is complete;
a. <i j . W. ( i j — G. ( i>
for all values of <x, therefore for all values of a.0, and there¬
fore W.0— I. _
Def .—In this case, either dyadic is called the reciprocal of
the other.
It is evident that an incomplete dyadic cannot have any
(finite) reciprocal.
Reciprocals of the same dyadic are equal. For if 0 and W
are both reciprocals of 12,
<b — <b.D. W = W.
If two dyadics are reciprocals, the operators formed by using
these dyadics as prefactors are inverse, also the operators formed
by using them as postfactors.
126. The reciprocal of any complete dyadic
aX + p jj. + y Y
is AV+ ///?'+ v’y’,
'where a\ ft', y' are the reciprocals of «, /?, p, and X\ ji , v are
the reciprocals of /, //, >. (See No. 38.)
VECTOR ANALYSIS,
47
127. Def *—We shall write 0~ x for the reciprocal of any
(complete) dyadic 0, also 0 2 for 0.0, etc., and 0)~ 2 , for
0~ 1 .0~ 1 , etc. It is evident that 0~ n is the reciprocal of 0 n .
128. In the reduction of equations, if we have
<i>. W = v.,t),
we may cancel the 0 (which is equivalent to multiplying by
<P~ r ) if 0> is a complete dyadic, hut not otherwise. The case is
the same with such equations as
a — #. p, w. 0 = £1.0, p.0— a. 0.
4
To cancel an incomplete dyadic in such cases would lie analo¬
gous to cancelling a zero factor in algebra.
129. T)ef .—If in any dyadic we transpose the factors in each
term, the dvadic thus formed is said to he conjugate to the first.
Thus
u A + (1 p + yv an d A a -f- pfi + vy
are conjugate to each other. A dyadic of which the value is
not altered by such transposition is said to he self-conjugate.
The conjugate of any dyadic 0 may he written 0 C 1 It is evi¬
dent that
p.a>—(i) c .p and r i\p — p.'/y.
0 c .p and 0 .p are conjugate functions of p. (See Ho. 106).
Since { 0 C } 2 — j 0 2 \c, we may write etc,, without ambi-
guity.
130. The reciprocal of the product of any number of dyadics
is equal to the product of their reciprocals taken in inverse
order. Thus
f\W.£lf'= £l-KW-£*-K
The conjugate of the product of any number of dyadics is
equal to the product of their conjugates taken in inverse order.
Thus
j *.w.n) c = n c ,w c .* c .
Hence, since
and we may write 0 ^ 1 without ambiguity.
131. It is sometimes convenient to he able to express by a
dyadic taken in direct multiplication the same operation which
would be effected by a given vector ( a ) in skew multiplication.
The dyadic I X« will answer this purpose. For, by Ho. 117,
18
VECTOR ANALYSIS.
{I Xa\.p =axp , p.|lx«'[ = p Xot,
{IX a }. <P —a X c b. [I X a\ — X a.
The same is true of the dyadic aX I, which is indeed identical
with 1 X a, as appears from the equation I .{aX 1} = {IX a}.1.
If a is a unit vector,
|IX«} 2 = — {I— fttx],
[IXo'| 8 = —I X a,
{I X o'} 4 — I—cm,
{I X o' | r> = IX a,
etc.
If /, j, h are a normal system of unit vectors,
IX* = ixi = [j-jk,
IX j = j X I = ik—hi ,
lXkz=zkxl— ji—ij •
If a and ft are any vectors,
[<*X/f] X I = I X [aX(3] — [3 a —a: [3.
That is, the vector ax[3 as a pre- or post-factor in skew mul¬
tiplication is equivalent to the dyadic \fta—aft] taken as pre-
or post-factor in direct multiplication.
[a X ft] X p = | (3a—a[3 } .p,
p X [a X /?] = p. | (3a— a(3 1.
This is essentially the theorem of No. 2ft, expressed in a form
more symmetrical, and more easily remembered.
132. The equation
a (3 X y h (3 y X a + y a X (3 — a. (3 X y I
gives, on mnltiplication by any vector p, the identical equation
p.a (3xy + p-(3 y X a p.y a X [3 — a.(3 Xy p.
(See No. 37.) The former equation is therefore identically
true. (See No. 108.) It is a little more general than the
equation
aa A- [3(3' y yy' — I,
which we have already considered (No. 124), since, in the form
here given, it is not necessary that a, ft, and y should be non-
complanar. We may also write
VECTOR ANALYSIS.
fiXy at + yxot fi + axfi y — oL.fi x y I.
Multiplying this equation by p as prefactor, (or the first equa¬
tion by p as postfactor,) we obtain
ft. fi X V a + ft. y X ol fi + ft.a x fi y — ol. fix y ft.
(Compare No. 87.) For three complanar vljjftors we have
ol fix y + fi y X ol + y a X fi = 0 .
Multiplying this by v, a unit normal to the plane of «, fft and
y, we have
ol fi X y. y + fi yXoi. v + y aX fi. v = 0.
This equation expresses the well-known theorem that if the
geometrical sum of three vectors is zero, the magnitude of
each vector is proportional to the sine of the angle between the
other two. It also indicates the numerical coefficients by
which one of three complanar vectors may be expressed in
parts of the other two.
183. T)ef .—If two dyadics 0 and 0' are such that
0. ¥ = W.4>,
they are said to be homologous.
If any number of dyadics are homologous to one another,
and any other dyadics are formed from them by the operations
of taking multiples, sums, differences, powers, reciprocals, or
products, such dyadics will be homologous to each other and
to the original dyadics. This requires demonstration only in
regard to reciprocals. Now if
0.W — W.&,
W. 0- 1 == 4 * - 1 .0. W. 0 1 =Z0- 1 .W.0.0~ 1 = 0^ 1 . W.
That is, is homologous to W, if 0 is.
IB!. If we call ¥.0 _1 or 0~ 1 .W the quotient of W and <P,
we may say that the rules of addition, subtraction, multiplica¬
tion and division of homologous dyadics are identical with
those of arithmetic or ordinary algebra, except that limitations
analogous to those respecting zero in algebra must be observed
with respect to all incomplete dyadics.
It follows that the algebraic and higher analysis of homol¬
ogous dyadics is substantially identical with that of scalars.
135. It is always possible to express a dyadic in three terms,
so that both the antecedents and the consequents shall be per¬
pendicular among themselves.
To show this for any dyadic 0, let us set
ft 7
ft’~'T>.ft,
50
VECTOR ANA LYSIS.
p being a unit-vector, and consider the different values of p'
for all possible directions of p. Let tlie direction of the unit
vector i be so determined that, when p coincides with b, tlie
value of [>' shall be at least as great as‘for any other direction
of p. And let the direction of the unit vector j be so deter¬
mined that when p coincides with j % the value of p' shall be at
least as great as for any other direction of p which is perpen¬
dicular to b. Let k liave its usual position with respect to i
and /'. It is evidently possible to express d> in the form
We have therefore
and
o' i -)- fij -(- yk.
p ~ J on -f- fij -f- yk j. ft,
dft — j cci 4- fij -)- yk \ Aft.
.Now the supposed property of the direction of i requires that
when p coincides with b and dp is perpendicular to /, dp' shall
be perpendicular to p', which will then be parallel to a. But
if dp is parallel to j or k, it will be perpendicular to i, and dp'
will be parallel to ft or y, as the case may be. Therefore ft and
y are perpendicular to a. In the same way it may be shown
that the condition relative to j requires that y shall be perpen¬
dicular to ft. IVe may therefore set
f i> z=z ai'i -f bfj -f- ck'k,
where bf jj kj like b, j, k, constitute a normal system of unit
vectors (see No. 11), and a, b. c are scalars which may be either
positive or negative.
It makes an important difference whether the number of
these scalars which are negative is even or odd. If two are
negative, say a and b , we may make them positive by reversing
the directions of i ! and j'. The vectors bj jj k! will still con¬
stitute a normal system. But if we should reverse the direc¬
tions of an odd number of these vectors, they would cease to
constitute a normal system, and to be superposable upon the
system i,j, k. We may, however, always set either
or
*T> — ai'i + bfj + ck’k,
<T> —: — { ai'i -|- bfj 4- ck'k },
with positive values of a, b, and c. At the limit between
these cases are the planar dyadics, in which one of the three
terms vanishes, and the dyadic reduces to the form
ai’i + bj'j,
in which a and b may always be made positive by giving the
proper directions to b' and j'.
VECTOR ANALYSIS.
51
If the numerical values of a, b, c are all unequal, there will
be only one way in which the value of 0 may be thus expressed.
If they are not all unequal, there will be an infinite number of
ways in which 0 may be thus expressed, in all of which the
three scalar coefficients will have the same values with excep¬
tion of the changes of signs mentioned above. If the three
values are numerically identical, we may give to either system
of normal vectors an arbitrary position.
136. It follows that any self-con jugate dyadic may be ex¬
pressed in the form
ail + bjj + ckk.
where i, j, k are a normal system of unit vectors, and a, b, c are
positive or negative scalars.
137. Any dyadic may be divided into two parts, of which
one shall be self-conjugate, and the other of the form I X«.
These parts are found by taking half the sum and half the
difference of the dyadic and its conjugate. It is evident that
<i> = £{<!>+ (I) c ! + 1 : rI) ~ % } •
low 1 j 0 -f- (J\ : j is self-con jugate, and
W 11 — <2> 0 } = Ix[--£* J|.
(See No. 131.)
Rotations and Strains.
138. To illustrate the use of dyadics as operators, let us sup¬
pose that a body receives such a displacement that
P' = &.p,
[> and p' being the position-vectors of the same point of the
body in its initial and subsequent positions. The same relation
will hold of the vectors which unite any two points of the
body in. their initial and subsequent positions. For if <> v p 2 are
the original position-vectors of the points, and />/, <p their
final position-vectors, we have
Pi —P 1? P'2 ~~ AP;j 5
whence
P'2 Pi -^'[Ps PlJ*
In the most general case, the body is said to receive a homo¬
geneous strain. In special cases, the displacement reduces to
a rotation. Lines in the body initially straight and parallel
will be straight and parallel after the displacement, and sur¬
faces initially plane and parallel will be plane and parallel
after the displacement.
52
VECTOR ANALYSIS.
139. The vectors (a, a') which represent any plane surface in
the body in its initial and final positions will be linear func¬
tions of each other. (This will appear, if we consider the four
sides of a tetrahedron in the body.) To find the relation of
the dyadics which express a' as a function of <7, and // as a
function of p, let
p — ■; a\ + ftp + Y v .}. p.
Then, if we write //, 1 / for the reciprocals of /, p, v, the
vectors //, F become by the strain «, /9, y. Therefore the
surfaces //.'xV, vxX, A'Xp' become /9 Xy, yXa , a X fi. But
P-'X'X, y'xX, A X [J. are ithe reciprocals of p X v, v x /, / X p.
The relation sought is therefore
< 3 ’ =4 ft x y M x y + y X ot v x A + a >&/i A X p }. cr.
140. The volume X'.p'Xv' becomes by the strain a.fixy.
The unit of volume becomes therefore (a. ft x y) {A.pXv).
—Tt follows that the scalar product of the three ante¬
cedents multiplied by the scalar product of the three conse¬
quents of a dyadic expressed as a trinomial is independent of
the' particular form in which the dyadic is thus expressed.
This quantity is the determinant of the coefficients of the nine
terms of the form
aii -f bij + etc.,
into which the dyadic may be expanded. We shall call it the
determinant of the dyadic, and shall denote it by the notation
S 1 /
\<P\
when the dyadic is expressed by a single letter.
If a dyadic is incomplete, its determinant is zero, and con¬
versely.
iS
The determinant of the product of any number of dyadics
is equal to the product of their determinants. The determi¬
nant of the reciprocal of a dyadic is the reciprocal of the deter¬
minant of that dyadic. The determinants of a dyadic and its
conjugate are equal.
The relation of the surfaces o’Xo may be expressed by the
equation
a' — |<i> | ^c" 1 cr.
141. Let us now consider the different cases of rotation and
strain as determined by the nature of the dyadic ( Jk
If is reducible to the form
i,j, /•, j\ k! being normal systems of unit vectors (see JNTo.
11), the body will suffer no change of form. For if
VECTOR ANALYSIS.
53
p = xi+yj+zk,
we shall have
pi = xi' + yf 4 - zk'.
Conversely, if the body suffers no change of form, the opera¬
ting dyadic is reducible to the above form. In snch cases, it
appears from simple geometrical considerations that the dis¬
placement of the body may be produced by a rotation about a
certain axis. A dyadic reducible to the form
i'i+jj+h'k
may therefore be called a versor.
142. The conjugate operator evidently produces the reverse
rotation. A versor, therefore, is the reciprocal of its conjugate.
Conversely, if a dyadic is the reciprocal of its conjugate, it is
either a versor, or a versor multiplied by —1. For the dyadic
may be expressed in the form
ai + fij+yk.
Its conjugate will be
iaA-j/3 + ky.
If these are reciprocals, we have
{oti -|- fij -f yk } . { la +j ft + ky | = aa + ftp 4- yy — I.
But this relation cannot subsist unless a, /?, y are reciprocals to
themselves, i. e., unless they are mutually perpendicular unit-
vectors. Therefore, they either are a normal system of unit-
vectors, or will become such if their directions are reversed.
Therefore, one of the dyadics
al 4- fij 4 - yk an d — ai — fij — yk
is a versor.
The criterion of a versor may therefore be written :
f J>. f l J c — I, and | ® | =1.
For the last equation we may substitute
i<C >0, or ].
It is evident that the resultant of successive finite rotations
is obtained by multiplication of the versors.
143. If we take the axis of the rotation for the direction of
i, i' will have the same direction, and the versor reduces to
the form
ii+j'j + k'k,
in which i, j, k and •/, j', k! a,re normal systems of unit vectors.
8
5 §
VECTOR AISTALYSIS.
«
We may set
/ = cos qj + Iu ^ k,
k' — cos q k — sin qj ,
and the versor reduces to
or
li + cos q\ jj+ kk\ -f sin q { hj —jit },
ii -f-cos q [I— ii } + sin q I x i,
o*£
v<-
«r-M rv\
cX*~r-
where q is the angle of rotation, measured from j toward k, if
the versor is used as a prefactor.
144. When any versor & is nsed as prefactor, the vector
— 0x will be parallel to the axis of rotation, and equal in
magnitude to twice the sine of the angle of rotation measured
counter-clock-wise as seen from the direction in which the
vector points. (This will appear if we suppose 0 to be repre¬
sented in the form given in the last paragraph.) The scalar
0 B will be equal to unity increased by twice the cosine of the
same angle. Together, — 0 x and 0 B determine the versor
without ambiguity. If we set
0 —
the magnitude of 0 will be
—
1 -f (? s 5
2 sin q
2 + 2 cos<?
or tan \ q,
t
where q is measured counter-clock-wise as seen from the direc¬
tion in which 0 points. This vector d, which we may call the
vector semitangent of version , determines the versor without
ambiguity.
145. The versor 0 may be expressed in terms of 6 in various
ways. Since 0 (as prefactor) changes a—dXa into a+Oxa
(a being any vector), we have
00 + jl + lx<9'f s _ (1—0.0)1 + 200+21x0
1+0.0 1+676 :
as will be evident on considering separately in the expression
0.p the components perpendicular and parallel to 0, or on sub¬
stituting in
ii + cos q ( jj + kit) + sin q ( kj—jk )
for cos q and sin q their values in terms of tan \ q.
If we set, in either of these equations,
0 = ai+bj+ck,
VECTOR ANALYSIS.
55
we obtain, on reduction, tbe formula,
(1 + u 3 — b 2 — c 2 )ii + (2ab—2c)ij+ (2ac + 2b)ik
-f (2ab-\-2c)ji+ (1— a 2 + b 2 —c 2 )jj + (2bc—2a)]k
+ {2ac—2b)M+{2bc + 2a)kj + ( \—a 2 — b 3 + c 2 )kk
1+u 3 +5 3 +c 3
in which the versor is expressed in terms of the rectangular
components of the vector semitangent of version.
146. If a, ft, y are unit vectors, expressions of the form
2aa—\ 2(3(3 —I, 2yy —I,
are biquadrantal versors. A product like
[2(3(3—\). [2 a a—1]
is a versor of which the axis is perpendicular to a and /9, and
the amount of rotation twice that which would carry a to /9.
It is evident that any versor may be thus expressed, and that
either a or j3 may be given any direction perpendicular to the
axis of rotation. If
<t> z=z[ 2(3(3—l}.[2aa— I}, and *P = [2yy—I}.[2/3fi— If,
we have for the resultant of the successive rotations
= { 2yy—I].[2aa—I]-.
This may be applied to the composition of any two successive
rotations, /? being taken perpendicular to the two axes of
rotation, and affords the means of determining the resultant
rotation by construction on the surface of a sphere. It also
furnishes a simple method of finding the relations of the vector
semitangents of version for the versors 0, I) and ¥. 0. Let
*i =
Then, since
-0 X - 0 x _J !p ® l V
_A Q — _ _2 Q _ l * • Z± *
l-f'/y 3 i + yy 3
0 = 4 a.(3 (3a — 2 aa — 2(3(3 + I,
f! __ a X(3
1 a.(3 ’
which is moreover geometrically evident. In like manner,
Therefore,
fiXy
a.(i ’
S _ a Xy
3 ~ a.y •
6 x u - [ g Xfl X [fiXy] __ ax(3.y (3
1 3 a. (3 (3.y a. (3 (3.y
— P' a P • X Y + P’P V X a + ( 3.y ax (3
a. (3 (3.y
56
VECTOR ANALYSIS.
(See Ho. 38.) Tliat is,
Also,
a. ft (3.y‘
a.y
Hence
A — A — ( 1_ A’A)A + ®±->
ft _ A + A + A X A
' Q ^ -
1-^.0
which is the formula for the composition of successive finite
rotations by means of their vector semitangents of version.
147. The versors just described constitute a particular class
under the more general form
a a’ -j- cos q{ fifi' + yy'} + sin q\ y/3 ! —j3y'},
in which «, /?, y are any non-complanar vectors, and /T, y'
their reciprocals. A dyadic of this form as a prefactor does
not affect any vector parallel to a. Its effect on a vector in the
ft-y plane will be best understood if we imagine an ellipse to
be described of which /3 and y are conjugate semi-diameters. If
the vector to be operated on he a radius of this ellipse, we may
evidently regard the ellipse with ;3, y, and the other vector, as
the projections of a circle with two perpendicular radii and one
other radius. A little consideration will show that if the third
radius of the circle is advanced an angle q, its projection in the
ellipse will be advanced as required by the dyadic prefactor.
The effect, therefore, of such a prefactor on a vector in the (3-y
plane may be obtained as follows :—Describe an ellipse of
which /? and y are conjugate semi-diameters. Then describe a
similar and similarly placed ellipse of which the vector to he
operated on is a radius. The effect of the operator is to
advance the radius in this ellipse, in the angular direction from
ft toward y, over a segment which is to the total area of the
ellipse as q is to 2 it. When used as a postfactor, the proper¬
ties of the dyadic are similar, hut the axis of no motion and the
planes of rotation are in general different.
Def .—Such dyadics we shall call cyclic.
The Hth power (N being any whole number) of such a
dyadic is obtained by multiplying q by H. If q is of the form
27rH/M (N and M being any whole numbers) the Mth power of
the dyadic will be an idemfactor. A cyclic dyadic, therefore,
may be regarded as a root of I, or at least capable of expression
with any required degree of accuracy as a root of I.
VECTOR ANALYSIS.
It should be observed that tlie value of the above dyadic
will not be altered by the substitution for a of any other
parallel vector, or for /? and y of any other conjugate semi-
diameters (which succeed one another in the same angular
direction) of the same or any similar and similarly situated
ellipse, with the changes which these substitutions require in
the values of a', /?', y'. Or, to consider the same changes from
another point of view, the value of the dyadic will not be
altered by the substitution for a! of any other parallel vector
or for [j and y' of any other conjugate semi-diameters (which
succeed one another in the same angular direction) of the same
or any similar and similarly situated ellipse, with the changes
which these substitutions require in the values of a , /?, and f,
defined as reciprocals of a', f, y'.
148. The strain represented by the equation
p 1 = {aii -f- bjj 4- okJc }. p
where a, b, c are positive scalars, may be described as consisting
of three elongations (or contractions) parallel to the axes i, j, k,
which are called the principal axes of the strain , and which
have the property that their directions are not affected by the
strain. The scalars a, b, c are called the principal ratios of
elongation. (When one of these is less than unity, it repre¬
sents a contraction.) The ordgr of the three elongations is
immaterial, since the original dyadic is equal to the product of
the three dyadics
aii-j-jj 4- kk, ii-\-bjj + ii-j- jjfckk
taken in any order.
Def .—A dyadic which is reducible to this form we shall
call a right tensor. The displacement represented by a right
tensor is called & pure strain. A right tensor is evidently self¬
conjugate.
149. We have seen (No. 135) that every dyadic may be
expressed in the form
=b | cti'i 4- bj'j -f ck'k },
where a, b , c are positive scalars. This is equivalent to
dr { ai'i' + bj'j'-\-ck'k '}. { i’ipj'j+k'k |
and to
d= { i'if-j'j + k'k |. {aii -f bjgfckh }.
Hence every dyadic may be expressed as the product of a
versor and a right tensor with the scalar factor ±1. The
versor may precede or follow. It will be the same versor in
either case, and the ratios of elongation will be the same ; but
58
VECTOR ANALYSIS.
the position of the principal axes of the tensor will differ in
the two cases, either system being derived from the other by
multiplication by the versor.
Def .—The displacement represented by the equation
P f = ~P
is called inversion. The most general case of a homogeneous
strain may therefore be produced by a pure strain ancl a rota¬
tion with or without inversion.
150. If
(p — ai'i -f bj'j -j- ck'k,
+ b 2 j'j' + cattle',
and f P c . <I> — a 2 ii -(- b 2 jj -)- c 2 kk.
The general problem of the determination of the principal
ratios and axes of strain for a given dyadic may thus be
reduced to the case of a right tensor.
151. Def .—The effect of a prefactor of the form
aaa' bftft - J- cyy 1
where a, b, c are positive or negative scalars, a, ft y non-com-
planar vectors, and o!, /J 7 , y' their reciprocals, is to change a.
into aa, ft into bft, and y into cy. As a postfactor, the same
dyadic will change a! into cut!, ft' into bft, and y r into cy!
Dvadics which can be reduced to this form we shall call tonic
-* J _
(dr. reivio). The right tensor already described constitutes a
particular case, distinguished by perpendicular axes and positive
values of the coefficients a, b, c.
The value of the dyadic is evidently not affected by sub¬
stituting vectors of different lengths but the same or opposite
directions for a, ft y, with the necessary changes in the values
of o!, ft, y ', defined as reciprocals of a, ft y. But, except this
change, if a, b, c are unequal, the dyadic can be expressed
only in one way in the above form. If, however, two of these
coefficients are equal, say a and b, any two non-collinear vectors
in the a-ft plane may be substituted for a and ft or, if the three
coefficients are equal, any three non-complanar vectors may be
substituted for a, ft y.
152. Tonics having the same axes (determined by the direc¬
tions of a , ft y) are homologous, and their multiplication is
effected by multiplying their coefficients. Thus,
{ aa’+b t fij3'+c 1 ^}. { a 2 aa , -\-b^/30' + c 2 yy'}
= { a % a 2 aa'-\- b ± b % e ± c 2 y y ' }.
Hence, division of such dyadics is effected by division of their
coefficients. A tonic of which the three coefficients a, b, c are
/
VECTOR ANALYSIS.
59
unequal, is homologous only with such dyadics as can he
obtained by varying the coefficients.
153. The effect of a prefactor of the form
aaa' + b{/3fd'-\-yy'\ + G {yfi'—/3y'},
or aaa' ~j -p cos q{/3fi'-\-yy'} -f -p sin q{yfi'—fiy'},
where a!, j3\ y' are the reciprocals of a, j3, y, and a, b, c, p, and
q are scalars, of which p is positive, will be most evident if we
resolve it into the factors
aaa’ -j- (3(3' -j- yy r ,
a a + p/3 (3' -\-pyy',
a a' U- cos q \ (3 (3'-\-yy' } + sin q { y ft—fly '},
of which the order is immaterial, and if we suppose the vector
on which we operate to be resolved into two factors, one
parallel to «, and the other in the {3-y plane. The effect of the
first factor is to multiply by a the component parallel to a,
without affecting the other. ’ The effect of the second is to
multiply by p the component in the [3-y plane without affecting
the other. The effect of the third is to give the component in
the ftp plane the kind of elliptic rotation described in No. 147.
The effect of the same dyadic as a postfactor is of the same
nature.
The value of the dyadic is not affected by the substitution
for a of. another vector having the same direction, nor by the
substitution for /? and y of two other conjugate semi-diameters
of the same or a similar and similarly situated ellipse, and
which follow one another in the same angular direction.
Def. —Such dyadics we shall call cyclotonic .
154. Cyclotonics which are reducible to the same form
except with respect to the values of a, p , and q are homolo¬
gous. They are multiplied by multiplying the values of a,
and also those of p, and adding those of q. Thus, the product
of
a t aa' +p 1 cosq i {/?/?' + yy'\ + p 1 smq 1 {y/3'—fjy'}
and a 2 aa' +p 2 cos q z + yy '\ + p 2 smq 2 {yf3'-f3y'}
is a t a 2 aa'^-p 1 p^cos{q t + q 9 ) j (3(3' + yy' }
+p 1 p^n(q i yq 2 ){y{3'-(3y'}.
A dyadic of this form, in which the value of q is not zero,
or the product of tt and a positive or negative integer, is homo¬
logous only with such dyadics as are obtained by varying the
values of a, p , and q.
155. In general, any dyadic may be reduced to the form
either of a tonic or of a cyclotonic. (The exceptions are such
60
VECTOR ANALYSIS.
as are made by the limiting cases.) We may show this, and
also indicate how the reduction may be made, as follows. Let
0 be any dyadic. We have first to show that there is at least
one direction of p for which
rp.p =. ap.
This equation is equivalent to
<P.p—ap — 0,
or, \$—dL).p = 0.
That is, (P=al is a planar dyadic, which may be expressed by
the equation
| 3 >— al\ — 0 .
(See No. 140). Let
(I> z=l A i + pj -f rk ;
the equation becomes
| [A— ai]i+[p—<xj\j + [v— ak~\k\ = 0,
or, [A— ai] X [p—aj] . [v— ah] — 0,
or,
a 3 — (i.X+j.p +i.v)a i + (i.pxr+j.i'>m. + k.\xp)a — hxp.r = 0 .
This may be written
a* — r / J s a 3 + { } s | ® \a — ! | 0.
Now if the dyadic (P is given in any form, the scalars
* 8 , |®|
are easily determined. We have therefore a cubic equation in
a, for which we can find at least one and perhaps three roots.
That is, we can find at least one value of a, and perhaps three,
which will satisfy the equation
|tf>—ai! = 0.
By substitution of such a value, 0-al becomes a planar dyadic,
the planes of which may be easily determined.* Let « be a
vector normal to the plane of the consequents. Then
{ <1> —«I|. a-— 0,
<I\a — aa.
If 0 is a tonic, we may obtain three equations of this kind,
say
* in particular cases, $—al may reduce to a linear dyadic, or to zero. These,
however, will present no difficulties to the student.
VECTOR ANALYSIS.
61
<T>. a — aa, <P.ft = bft, V.y = cy,
in which a, ft, y are not complanar. Hence, (by Ho. 108,)
<l > = + bftft' + cyy',
where a', ft', y' are the reciprocals of a, ft, y.
In any case, we may suppose a to have the same sign as 101,
since the cubic equation must have such a root. Let a (as
before) he normal to the plane of the consequents of the
planar (P—al, and a! normal to the plane of the antecedents,
the lengths of a. and a! being such that a.a! — 1 .* Let ft he any
vector normal to a!, and such that @.ft is not parallel to ft.
(The case in which <P.ft is always parallel to ft, if ft is perpen¬
dicular to a', is evidently that of a tonic, and needs no farther
discussion.) \<P—aI}.ft and therefore ®.ft will he perpendicu¬
lar to a! . The same will he true of @*.ft. How (by Ho. 140)
[0.«].[0 3 ./5]X = ! 0| a.[&./3] X ft,
that is,
aa.[®*.ft>] X[ ( I J -ft] = l®l a.[$.ft’]xft.
Hence, since [ f l j2 .ft]x[ ( f > .ft ] and [d>.ft]xft are parallel,
ci[<i>*.ft] X [*-ft] =101 [*.ft] X ft.
Since a~ 1 \@\ is positive, we may set
p 2 = cr 1 |'H
If we also set
ft t =p-i<P.ft, ft 2 =p~*$ 2 .ft, etc.,
ft_ x —p<D- i.ft, ft _ 2 Jk^V-z.ft, etc.,
the vectors ft, ft t , ft 2 , etc., ft_ t , etc., will all lie in the plane
perpendicular to a', and we shall have
ft 2 Xft t =ft t Xft,
[ft 2 + ft] X fti = o.
We may therefore set
ft % + ft = 2n ft t .
Multiplying byy? - 1 0, and by
ft 3 + ft 1 = 2 nft 2 , /i 4 + ft z = 2nft 3 , etc.,
ft 1 - 1 - ft„ t = 2 nft, ft + ft_ 2 = 2 nft_ 1 , etc.
How, if n> 1 , and we lay off from a common origin the vectors
ft > ftft etc., ft—t 5 ft— 2 ’ etc.,
* Eor the case in which the two planes are perpendicular to each other, see No.
157.
9
62
VECTOR ANALYSIS.
the broken line joining the termini of these vectors will be
convex toward the origin. All these vectors must therefore
lie between two limiting lines, which may be drawn from the
origin, and which may be described as having the directions of
/Soo and /?_oo .* A vector having either of these directions is
unaffected in direction by multiplication by 0. In this case,
therefore, 0 is a tonic. If n<— 1, we may obtain the same
result by considering the vectors
ft i ft it ft %t ft 3’ ft 4 ) etG -, ft— i< ft—Z’ ft—3’ G *tC.,
except that a vector in the limiting directions will be reversed
in direction by multiplication by 0, which implies that the
two corresponding coefficients of the tonic are negative.
If l,f we may set
Then
n = cos q.
P- t + = 2 cos q /?.
Let us now determine y by the equation
^ = cos q $ + sin q y.
This gives
(L-l = cos q§ — sin q y.
How a! is one of the reciprocals of a, /?, and y. Let ft and y'
be the others. If we set
W =. (iosq{P@' + yy r } 4- sin q\yft' —fty'\,
we have
W.a = 0, WJ = = ft
Therefore, since
{ aaa' +p W).a — aa — 0. n,
{aaa' -\-pW}.@ =p@ 1 = *P.ft
[aaa' +pW}.^_ t =p@ =
it follows (by Ho. 108) that
<P =z aaa' +pW = aaa' +p cos q{§§' + yy' } sin q{yft'—fty' }*
156. It will be sufficient to indicate (without demonstration)
the forms of dyadics which belong to the particular cases which
have been passed over in the preceding paragraph, so far as
they present any notable peculiarities.
* The termini of the vectors will in fact lie on a hyperbola,
f For the limiting cases, in which n— 1, or n= — 1, see No. 156.
VECTOR ANALYSIS.
63
If n= dtl, (page 62,) the dyadic may be reduced to the form
aaa + b{ftft' + yy'} + hefty',
where a, /3, y are three non-complanar vectors, a, (3', y 1 their
reciprocals, and a, b, c positive or negative scalars. The effect
of this as an operator, will be evident if we resolve it into the
three homologous factors
aaa' + ftft' + yy',
aa' + b{ftft' + yy'\,
aa 4- ftft' +yy' + cfty'.
The displacement due to the last factor may be called a simple
shear. It consists (when the dyadic is used as prefactor) of a
motion parallel to /3, and proportioned to the distance from the
a-j3 plane. This factor may be called a shearer.
This dyadic is homologous with such as are obtained by vary¬
ing the values of a , b, c, and only with such, when the values
of a and b are different, and that of c other than zero.
157. If the planar <P-al (page 61) has perpendicular planes,
there may be another value of a, of the same sign as | @\, which
will give a planar which has not perpendicular planes. When
this is not the case, the dyadic may always be reduced to the
form
a{aa'-j-ftft' + yy'} + ab{aft'+ ft y'} + acay',
where a, /3, y are three non-complanar vectors, a ', ft, y', their
reciprocals, and a , b, e, positive or negative scalars. This may
be resolved into the homologous factors
al and I -1- 5] aft'+fty'} + cay’.
The displacement due to the last factor may be called a complex
shear. It consists (when the dyadic is used as prefactor) of a
motion parallel to a which is proportional to the distance from
the a-y plane, together with a motion parallel to b[3 +• ca which
is proportional to the distance from the «-/3 plane. This factor
may be called a complex shearer.
This dyadic is homologous with such as are obtained by
varying the values of a, b, c, and only such, unless b=0 .
It is always possible to take three mutually perpendicular
vectors for a , /?, and y; or, if it be preferred, to take such
values for these vectors as shall make the term containing c
vanish.
158. The dyadics described in the two last paragraphs may
be called shearing dyadics.
64
VECTOR ANALYSIS.
The criterion of a shearer is
) I } 3 = 0 .
The criterion of a simple shearer is
jtf-Ifs = 0.
The criterion of a complex shearer is
j<Z>-I} 3 =0, > 0.
Note.—I f a dyadic 4> is a linear function of a vector p, (the term linear being
used in the same sense as in No. 105,) we may represent the relation by an
equation of the form
$ = a/3 y.p + eC y.p + etc.,
or 4> = afiy + e£>/ + etc. y . p,
where the expression in the braces may be called a triadic polynomial , and a
single term a(3y a triad , or the indeterminate product of the three vectors a, /3, y.
We are thus led successively to the consideration of higher orders of indeter¬
minate products of vectors, triads , tetrads , etc., in general polyads, and of polyno¬
mials consisting of such terms, triadics, teiradics, etc., in general polyadics. But
the development of the subject in this direction lies beyond our present purpose.
It may sometimes be convenient to use notations like
P, v - h P, v
h P,y P, rl
to represent the conjugate dyadics which, the first as prefactor, and the second
as postfactor, change a, /3, y into A, y, v, respectively. In the notations of the
preceding chapter these would be written
la' + pj3' + vy' and a'% + (i'p + y'v
respectively, a', fi', y' denoting the reciprocals of a. (3, y. If r is a linear function
of p, the dyadics which as prefactor and postfactor change p into t may be
written respectively
i —■ and —7.
Ip pI
If r is any function of p, the dyadics which as prefactor and postfactor change dp
into dr may be written respectively
dr
\dp
and
dr
In the notation of the following chapter the second of these, (when p denotes a posi¬
tion-vector), would be written yr. The triadic which as prefactor changes dp into
cl) d 2 T d
.—- may be written .—and that which as postfactor changes dp into —may be
| dp I dp 2 dp |
d 2 t
written ——n. The latter would be written yyr in the notations of the following
dp* |
chapter.
VECTOR ANALYSIS.
65
CHAPTER IY.
[Supplementary to Chapter II.]
CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS
OF VECTORS.
159. If a) is a vector having continuously varying values in
space, and p the vector determining the position of a point, we
may set
p — x i + yj + z A
dp — dx i + dyj + dz />;,
and regard to as a function of />, or of x , y, and v. Then,
that is.
If we set
7 , doo , g?go T t?co
«g» — cm: — + ——(- dz —,
tviK ■'$/ Ci'Z
, , ( . c?g» Ago t Jgo )
dm = dp. j »-r- +.?-*- + * — <■
. cho
Jco
C«fcO
Here /7 stands for
G^GO = dp.\/GD.
. d , d 7 d
G?£C
Gfe’
exactly as in Ho. 52, except that it is here applied to a vector
and produces a dyadic, while in the former case it was applied
to a scalar and produced a vector. The dyadic pto represents
the nine differential coefficients of the three components of to
with respect to x, y , and z, just as the vector yu (where u is a
scalar function of p) represents the three differential coefficients
of the scalar u with respect to x, y, and z.
It is evident that the expressions p.to and yX to already
defined (Ho. 54), are equivalent to jj7Goj s and \(7to\ x .
66
VECTOR ANALYSIS.
160. An important case is that in which the vector operated
on is of the form pit,. We have then
»
dpu = dp.ppu,
where
d?
u
dx‘
ii +
d 2 u .. d*u ... j
-y _(_ _ _ J
dxdy
d^u
dho ..
rr*=< + 3 - Jx p + dy
-33 +
dxdz
d 2 u .
dydti
ik
ft )■
d*u , . d*u T . d 2 ii 7 7
-)——— —kb -f- — 7 —— kj -|- — kk.
dzdx
dzdy
d?d
J
This dyadic, which is evidently self-con jugate, represents the
six differential coefficients of the second order of u with respect
to a?, y, and a*
161. The operators pX and p. may be applied to dyadics in
a manner entirely analogous to their use with scalars. Thus
we may define pX <P and p.(P by the equations
Then, if
Z t i
Or, if
px$
. d ( i j . dd> d ( J j
FX lX Tx +jX ~dv + kX li
(vih \AJ LJ \Aj&
. d ( i j . d<T> 7 d$
r-'” = l -7f x +J -Ty + h ^-
<I> — a i pj + y k,
p-X®=pXai + p X P j + pXyk,
p.V= p.ai +p.8 y + p.yk.
<l>— ia +jB + k y,
dy dp
dy dz
+j
da dy
dz dx
+ k
dp
dx
da dp dy
1 — dx ^ dy dz'
da
dy
162 We may now regard p.p in expressions like p.pto as
representing two successive operations, the result of which
will be
d 2 oi d 2 co d‘ 2 oi
dx 2 dy* dz 2
in accordance with the definition of No. 70. We may also
write p.p^ for
* We might proceed to higher steps in differentiation by means of the triadics
W«, vW w i the tetradics vw°! VVVV M t etc - See note on page 64. In like
manner a dyadic function o£ position in space (<J>) might he differentiated by means
of the triadic y<I>. the tetradic yy<t», etc.
VECTOR ANALYSIS.
67
<r ® <p<i> d*®
!h? + df + d?r
although, in this case we cannot regard p.p as representing two
successive operations until we have defined p0f
That p.p(P=pp.<P—pXpX <P will he evident if we suppose
0 to he expressed in the form oh4- ftj-Pjh. (See hTo. 71.)
163. We have already seen that
u"—vl —f dp.pu ,
where u' and u" denote the values of u at the beginning and
the end of the line to which the integral relates. The same
relation will hold for a vector
%. e..
go" — go' =f dp. poo.
164. The following equations between surface-integrals for a
closed surface and volume-integrals for the space enclosed seem
worthy of mention. One or two have, already been given, and
are here repeated for the sake of comparison.
ffda u —fffdv pip
(i
ff da GD —fffdv pGD,
( 2 )
/ ffda.GD -fffdv p.ao,
00
f ff da.® = fffdv p.®.
(0
V /7 dax go = fffdv px go,
( 5 )
ffda X ® =fff dv px®.
( 6 )
X
It may aid the memory of the student to observe that the
transformation may be effected in each case by substituting
fffdv? for //do,.
165. The following equations between line-integrals for a
closed line and surface-integrals for any surface bounded by
the line, may also be mentioned. (One of these has already
been given. See hTo. 60.)
f dpu — ff daxpu, ( 1 )' ~
fdp go =ff daxpoo, ( 2 )
fdp.GD=f/dcr.pxoj, (3)-"£<TF =
fdp.&=ffda.px@, ( 4 ) *=
j dp X go =ffp GD.dc> -ffdap. ca. ( 5 )^ y T/ _ _ ^
These transformations jnay be effected by substituting _ ^
ff\daxp] for ffdp. The brackets are here introduced
to indicate that the multiplication of da with the 7, j, k
implied in p is to be performed before any other multiplica-
* See foot-note to No. 160.
V
; _ ^Vrv.J ~ £ V .Vtyv.A/-JP
— XI (vX — /V./Vv- 1 ?) X« (
68
VECTOR ANALYSIS.
tion which, may he required hy a subsequent sign. (This
notation is not recommended for ordinary use, hut only sug¬
gested as a mnemonic artifice.)
166. To the equations in No. 65 may be added many others,
as,
/7[Mchj = f7U GJ + up GJ, (1)
F[ix go] = [7TX GJ — 17&>XT, (2)
|7X[lX©] = GJ.f/t — J7.r GJ — T. 17GJ + fr.GJT , (3)
pr(r.Gj) = (7L GJ + [7CJ.T, (4)
p'.{TGj}= 17.T GJ + T.£7GJ, (5)
[7X {tgj}= [7XT GJ — TX{7GJ, (6)
£7.{u$}— 1711 .$ + up.&, (7)
etc.
The principle in all these cases is that if we have one of the
operators p, p., pX prefixed to a product of any kind, and we
make any transformation of the expression which would he
allowable if the p were a vector, (viz : hy changes in the order
of the factors, in the signs of multiplication, in the parentheses
written or implied, etc.,) hy which changes the p is brought
into connection with one particular factor, the expression thus
transformed will represent the paid of the value of the original
expression which results from the variation of that factor.
161. From the relations indicated in the last four para¬
graphs, may he obtained directly a great number of trans¬
formations of definite integrals similar to those given in Nos.
14-71, and corresponding to those known in the scalar calculus
hy the name of integration by parts.
' 168. The student will now find no difficulty in generalizing
the integrations of differential equations given in Nos. 18-89
by applying to vectors those which relate to scalars, and to
dyadics those which relate to vectors.
169. The propositions in No. 90 relating to minimum values
of the volume-integral fff uco.codv may he generalized hy sub¬
stituting ( 0 . 0 .co for uco.a ;, 0 being a given dyadic function of
position in space.
110. The theory of the integrals which have been called
potentials, Newtonians, etc. (see Nos. 91-102) may be ex¬
tended to cases in which the operand is a vector instead of a
scalar or a dyadic instead of a vector. So far as the demon¬
strations are concerned, the case of a vector may be reduced to
that of a scalar by considering separately its three components,
and the case of a dyadic may be reduced to that of a vector,
by supposing the dyadic expressed in the form ipi+yj+cok and
considering each of these terms separately.
VECTOR ANALYSIS.
69
CHAPTER V.
CONCERNING TRANSCENDENTAL FUNCTIONS OF DYADTCS.
171. Def -—The exponential f unction, tlie sine and the
cosine of a dyadic may he defined by infinite series, exactly as
the corresponding functions in scalar analysis, viz :
These series are always convergent. For every value of 0
there is one and only one value of each of these functions.
The exponential function may also be defined as the limit of
the expression
when 1ST, which is a whole number, is increased indefinitely.
That this definition is equivalent to the preceding, will appear
if the expression is expanded by the binomial theorem, which
is evidently applicable in a case of this kind.
These functions of 0 are homologous with 0.
172. We may define the logarithm as the function which is
the inverse of the exponential, so that the equations
e*= 0,
W = log 0,
are equivalent, leaving it undetermined, for the present,
whether every dyadic has a logarithm, and whether a dyadic
can have more than one.
173. It follows at once from the second definition of the
exponential function that, if 0 and W are homologous,
and that, if T is a positive or negative whole number,
i f .t__ ae
10
TO
VECTOR ANALYSIS.
174. If E and ¥ are homologous dyadics, and sncli that
0 = — £>,
the definitions of Ho. 171 give immediately
— cos <I> + E sin <I>,
e~~-® z= cos 0 — E sin <I\
whence
cos 0 = £.{,
sin 0 =. — ^E{e A ®—
175. If 0.¥=¥.d>= 0,
I 0+ W} 2 = 0 2 + ¥\ \ 0+Wy= 0 s + ¥\ etc.
Therefore
«* + *=«* + e*-I,
cos | 0 + W }= COS 0 + cos W —I,
sin | 0-\- W}= sin 0 + sin W.
176.
For the first member of this equation is the limit of
Ijl + N-i^H, that is, of |I + N-i$| N .
If we set 0—aiE- ftj + jh, the limit becomes that of
(l+N _1 o , .«4-N“ 1 ^.J + N _1 y.^) Isr , or (l + N -1 ^)^,
the limit of which is the second member of the equation to be
proved.
177. By the definition of exponentials, the expression
e <i\ iq-jk y
represents the limit of
{I + q T$-i{kj-jk}}«.
How I + ^H -1 \hj—'jk\ evidently represents a versor having the
axis i and the infinitesimal angle of version pH -1 . Hence the
above exponential represents a versor having the same axis and
the angle of version q." If we set qi—co , the exponential may
be written
& I X 6)
Such an expression therefore represents a versor. The axis and
direction of rotation are determined bv the direction of to, and
«/ 7
VECTOR ANALYSIS.
71
the angle of rotation is equal to the magnitude of co. The
value of the versor will not be affected by Increasing or dimin¬
ishing the magnitude of oj by 2 tt.
178. If, as in Ho. 151,
0 = aaa' + bftft' 4- cyy',
the definitions of Ho. 171 give
e $ — e a aa ' _)_ e^ftft' + e c yy',
cos 0 = cos a aa' -f cos b ft ft' + cos c yy',
sin 0 — sin a aa! + sin b ft ft' + sin c yy'.
If a, b, c are positive and unequal, we may add, by Ho. 172,
log 0 — log a aa' + log b ft ft 1 + logc yy'.
179. If, as in Ho. 153,
0 — aaa' 4- b\ftft' + yy'\ + c{yft'—fty'}
— aaa' + p cos q{ftft' + yy'} + p si nq{yft'—fty f },
we have by Ho. 173
V
e®=e aaa \e h l ft ft'+ 77' 7ft'—ft7' 1 .
But
e <xaa' __ G tt aa ’ _J_ yy’
e b < PP'+yy' y=aa' + e^^ft' + yy'}
\ yft'—fty' Y — aa ’ _j_ cos c {ftft' + yy'}-\- sin c {yft'—fty' [.
Therefore,
e® — e a aa' -f- e 1 * cosc{ftft'-{-yy'}-\- sin c\ yft' — fty'}.
Hence, if a is positive,
log 0 = log a aa' -f- iogp{ftft'+yy'}-\- q_\yft—fty'}-
Since the value of 0 is not affected by increasing or dimin¬
ishing q by 2 tt, the function log 0 is many-valued.
To find the value of cos 0 and sin 0, let us set
© = b{ftft'-\-yy'}-yc{yft'—fty'},
'B = yft' — fty'-'
Then, by Ho. 175,
cos 0 = cos { aaa'} -J- cos © — I.
But
cos j aaa'} — I = cos a a a' — aa',
72
VECTOR ANALYSIS.
Therefore,
cos ft = cos a a eft — a a + cos ©.
Now, by No. 174,
cos & = 4 {e~- e +c~ s - 0 }.
Since
£.© =
e^- e = neb -f- e _c cosi{^'-|ly , } 4 - c~ c sin &{/(?'' — (fy'j,
e~ ~' e — at at' + e c cos ?) ( 00 ' + 77 '}— e c sin 5 { 7 ^' — ^ 7 '}.
Therefore
cos©=u'n' , -j- 4 (e c +e~ c ) cos — £(<? c —e ~ c )sin b{yfi'—fiy'\,
and
cos 0 = cos a ota'-{-%(e c -\-e G ) cos
— l(e c — e — c )$mb{y@ r —fty'}
In like manner we find
sin <P = sin a aa'-\-^(e c -\-e~ c ) sin
-f-l(e c —e~ c ) cos b{y8'—@y'\.
180. If a, ft , y and eft, ft', y' are reciprocals, and
ft = aaa' -|- b{@@'-\-yy'\ 4- cBy',
and N is any whole number,
< 2 > N = aftaoe 4 - b N \@@'-\-yy'} 4 - N b^^cBy'.
Therefore,
eft — e a a a' 4 - ft' {&§'ft-yy'}-{- ft'cfiy',
cos ft — cos a a a' + cos b{fi@'-\-yy'} — csiu b fty\
sin ft — sin a a a' -j- sin b{§ft-ftyy'} + c cos b fty'.
If a and b are unequal, and 0 other than zero, we may add
log ft = log a a a' -f log b{§ft-\-yy'\ft- dr 1 ^.
181. If a , ft, y, and eft, ft', y' are reciprocals, and
ft — al -f- b{a@'-\-py'\-{- cay',
and N is a whole number,
ft N = u N I + N« N -i b{ aft’ + ftft } + (N a N -i c + IN (N - 1 ) aft~^)ay'.
Therefore
VECTOR ANALYSIS.
73
e a l + e a b{aP'+Py'} + e a (ft*+c) ay',
cos <P — cos al—b sin a{ a@' + @a’} — {^b 2 cos a-{-c sin a) ay',
sin <P = sin a 1+5 cos a { aft + $ot! } — (^5 2 sin a—c cos a)ay'.
Unless 5=0, we may add
log f i> = log a I + bcr 1 { a@' + @a' \ -j- {car 1 —\b 2 ar 2 )ay'.
182. If we suppose any dyadic 0 to vary, but 'with the
limitation that all its values are homologous, we may obtain
from the definitions of Ho. 171
cl{e®\ = e ^. d*P,
(>)
d sfn <2> = cos CP . d<P ,
(2)
d cos 0 z=i —sin 0 . d0,
0)
dlog 0 =z (P^ 1 . d<i J ,
W
as in the ordinary calculus, but we must not apply these
equations to cases in which the values of 0 are not homologous.
183. If, however, F is any constant dyadic, the variations
of tl will necessarily be homologous with tf, and we may
write without other limitation than that T is constant,
d sinj£U|
dt
= F. cos {tF},
d cos | tF }
dt
= - F sin {tF}
d\og{tF } _I
dt ~ t ‘
(i)
(0
(0
(4)
A second differentiation gives
cP{e tv \ _ tT
rr
cP sin ;// ' ( _
dt 2
/A sini tl
) 3
d 2 cos{tF }
dt 2
-F*.cos{tF}.
(®)
(6)
(0
181. It follows that if we have a differential equation of
the form
dp
dt ~ * P ’
the integral equation will be of the form
74
VECTOR ANALYSIS.
p = e .p ,
,tv ^
p' representing the value of p for t= 0. For this gives
and the proper valne of p for t=0.
185. I)ef .—A flux which is a linear function of the position-
vector is called a homogeneous-strain-flux from the nature of
the strain which it produces. Such a flux may evidently be
represented by a dyadic.
In the equations of the last paragraph, we may suppose p to
represent a position-vector, t the time, and P a liomogeneous-
strain-flux. Then e tr will represent the strain produced by the
flu x F in the time t.
In like manner, if A represents a homogeneous strain,
\ log A\/t will represent a homogeneous-strain-flux which would
produce the strain A in the time t.
186. If we have
where / is complete, the integral equation will he of the form
p — e tl .a-\-e
For this gives
and a and /9 may be determined so as to satisfy the equations
P(=o=“ + ft
187. The differential equation
rl. 2 n
will be satisfied bv
t/
p = cos{tP}. a + sin {tr }. ft,
FA — - r. sin j tP) . a + P. cos \tP }. <?,
whence
VECTOR ANALYSIS.
75
d 2 p
dt 2
— F 2 . cos {tF }. a — F 2 . sin {tF \. (3 — —F 2 .p,
\ bL v
If /" is complete, the constants o. and ft may he determined to
satisfy the equations
P
t -o
O',
\F\ = r.§.
L dU t _ 0
188. If
d 2 p
dt 2
\F 2 —A 2 ), p,
where r 2 —A 2 is a complete dyadic, and
F.A = A.F= 0,
we may set
[i tr
-tv
tv
p=\^e + %e + cos {£ y 4 } — — 1
which gives
dp = U r J r _^F.e~ tV
-tv
fe +sin
dt
-A. sin{ t/1 } \. a
( tv —tv )
4 -\\F.e +\I\e + A. cos{ tA } \. &
dSp ={ir 2 .e T + ir*.e tT -A*.co8{tA\}.a
dt"-
( fp _/p )
+ \iF 2 .e -\F 2 .e ‘ -A 2 .sm{tA}\.8.
- j F 2 -A 2 \. p.
The constants a and ft are to be determined by
(A
— Of,
r dp
L dt —
t =o
= j F+Al.p.
t- o
189. It will appear, on reference to hi os. 155-157, that every
complete dyadic may be expressed in one of three forms, viz:
as a sqnare, as a sqnare with the negative sign, or as a differ¬
ence of squares of two dyadics of which both the direct pro¬
ducts are equal to zero. It follows that every equation of the
form ,
d 2 p
df* = e - p
where 0 is any constant and complete dyadic, may be inte¬
grated by the preceding formulae.
76
BIVECTOR ANALYSIS.
NOTE cm BIVECTOK ANALYSIS.*
1. A vector is determined by three algebraic quantities. It
often occurs that the solution of the equations by which these
are to be determined gives imaginary values; i. e., instead of
scalars we obtain biscalars, or expressions of the form a+eb,
where a and b are scalars, and i—\/ — 1. It is most simple,
and always allowable, to consider the vector as determined by
its components parallel to a normal system of axes. In other
words, a vector may be represented in the form
® i + yj + ^
Now if the vector is required to satisfy certain conditions, the
solution of the equations which determine the values of x, y,
and . 3 , in the most general case, will give results of the form
X — X 1 -\- l x. 2 ,
y = y i + i y
2 = «i 4-
where aq, rr 2 , y t , y 2 ,
values in
we obtain
z 1 , s 2 are scalars.
* 1 + yj +«
Substituting these
(x t +ix 2 )i- f- {y y + iy z )j+ («,4
or, if we set
* Thus far, in accordance with the purpose expressed in the foot-note on page
1, we have considered only real values of scalars and vectors. The object of this
limitation has been to present the subject in the most elementary manner. The
limitation is however often inconvenient, and does not allow the most symmetrical
and complete development of the subject in many important directions. Thus in
Chapter V, and the latter part of Chapter III, the exclusion of imaginary values
has involved a considerable sacrifice of simplicity both in the enunciation of
theorems and in their demonstration. The student will find an interesting and
profitable exercise in working over this part of the subject with the aid of
imaginary values, especially in the discussion of the imaginary roots of the cubic
equation on page 60, and in the use of the formula
e = cos $-M sin <P
in developing the properties of the sines\ cosines, and exponentials of dyadics.
BIVECTOR ANALYSIS.
< (
we obtain
p 1 =x 1 i + yj -f sq k,
p 2 = x 2 i + yj + 2„ k,
Pi 4 ~ L P%-
We shall call this a bivector, a term which will include a vector
as a particular case. When we wish to express a bivector by a
single letter, we shall use the small German letters. Thus we
may write
* — Pi + 1 Ps-
An important case is that in which y 1 and y 2 have the same
direction. The bivector may then be expressed in the form
(a+eb)p, in which the vector factor, if we choose, may be a
unit vector. In this case, we may say that the bivector has a
real direction . In fact, if we express the bivector in the form
Oh+#2 ) i + k
the ratios of the coefficients of i,j, and k, which determine the
direction cosines of the vector, will in this case be real.
2. The consideration that operations upon bivectors may be
regarded as operations upon their biscalar x- y- and .s-compo-
nents is sufficient to show the possibility of a bivector analysis
and to indicate what its rules must be. But this point of view
does not afford the most simple conception of the operations
which we have to perform upon bivectors. It is desirable that
the definitions of the fundamental operations should be inde¬
pendent of such extraneous considerations as any system of
axes.
The various signs of our analysis, when applied to bivectors,
may therefore be defined as follows; viz :
The bivector equation
y' ir' = fx" -f- iv"
implies the two vector equations
y! — y", and v' = r".
— [y+iv] — —y -f z[— v].
[y' + iv'] + [y"-\-ir"] - [// + //] + i[v'~ fV|j
[y' + iv'] . [y"+ir"] = [y'.y"~r'.r"] -f z [y'.v”+ r'.y"].
[y'+iv']x[y"+ LV "] = [p'Xy"-r T Xr"]+ i[y'xr"-\-r'xy"].
With these definitions, a great part of the laws of vector
analysis may be applied at once to bi vector expressions. But
an equation which is impossible in vector analysis may be pos¬
sible in. bivector analysis, and, in general, the number of roots
11
BIVECTOR ANALYSIS.
hr Q
i o
of an equation, or of the values of a function, will he different
according as we recognize, or do not recognize, imaginary
values.
3. Def .—Two bivectors, or two biscalars, are said to be con¬
jugate, when their real parts are the same, and their imaginary
parts differ in sign, and in sign only.
Hence, the product of the conjugates of any number of
bivectors and biscalars is the conjugate of the product of the
bivectors and biscalars. This is true of any kind of product.
The products of a vector and its conjugate are as follows:
[M 4 iv ]. \ju— iv] = }a+}a -f y.y
[fii+ ir] X \ fi~ir] — 2irXM
[M+ lv \ [A *— ir ] — \ MM+ vv\ -f- i\ r/A—pir\.
Hence, if /a and tv represent the real and imaginary parts of
a bivector, the values of
jA.ju+r.v, JuXu, /qn-j- w, y/x—jAV,
are not affected by multiplying the bivector by a biscalar of
the form a+tb, in which a 2 +b 2 = l. Thus, if we set
we shall have
f+ir 1 =z («-f ib)[jj- f- iv\
and
That is,
f.i' — iv' — [a — ib)\_fA — iv\
[q'+ — iv'\ = iv\.
ja'.ja' 4- r'.y' = /u/I-4- y.y ;
and so in the other cases.
4. Def .—In biscalar analysis, the product of a biscalar and its
conjugate is a positive scalar. The positive square root of this
scalar is called the modulus of the biscalar. In bivector analy¬
sis, the direct product of a bivector and its conjugate is, as
seen above, a positive scalar. The positive square root of this
scalar may be called the modulus of the bmector. When this
modulus vanishes, the bivector vanishes, and only in this case.
If the bivector is multiplied by a biscalar, its modulus is mul¬
tiplied by the modulus of the biscalar. The conjugate of a
(real) vector is the vector itself, and the modulus of the vector
is the same as its magnitude.
5. Def .—If between two vectors, a and f there subsists a
relation of the form
a = nS,
where n is a scalar, we say that the vectors are parallel.
BIYECTOE ANALYSIS.
T9
Analogy leads us to call two bivectors parallel, when there
subsists between them a relation of the form
a = mb,
where m (in the most general case) is a biscalar.
To aid us in comprehending the geometrical signification of
this relation, we may regard the biscalar as consisting of two
factors, one of which is a positive scalar, (the modulus of the
biscalar,) and the other may be put in the form cos q -f i sin q.
The effect of multiplying a bivector by a positive scalar is
obvious. To understand the effect of a multiplier of the form
cos q -f t sin q upon a bivector p+tv, let us set
p'+iv' = (cos q + i sin q)[p + iv\
We have then
// = cos q p — sin q v ,
v f = cos q r + sin q p.
How if p and v are of the same magnitude and at right angles,
the effect of the multiplication is evidently to rotate these
vectors in their plane an angular distance q, which is to be
measured in the direction from v to p. In any case we may
regard a and v as the projections (by parallel lines) of two per¬
pendicular vectors of the same length. The two last equations
show that // and v will be the projections of the vectors
obtained by the rotation of these perpendicular vectors in their
plane through the angle q. Hence, if we construct an ellipse
of which p and u are conjugate semi-diameters, p! and v' will
be another pair of conjugate semi-diameters, and the sectors
between p and p! and between u and i/, will each be to the
whole area of the ellipse as q to 2 tt, the sector between v and v f
lying on the same side of v as //, and that between p and p!
lying on the same side of p as — v.
It follows that any bi vector p+tv may be put in the form
(cos q 4- i sin q) [n-f- z(5],
in which a and J are at right angles, being the semi-axes of the
ellipse of which p and v are conjugate semi-diameters. This
ellipse we may call the directional ellipse of the bi vector. In
the case of a real vector, or of a vector having a real direction,
it reduces to a straight line. In any other case, the angular
direction from the imaginary to the real part of the bivector is
to be regarded as positive in the ellipse, and the specification
of the ellipse must be considered incomplete without the indi¬
cation of this direction.
Parallelism of bivectors, then, signifies the similarity and
80
BIVECTOR ANALYSIS.
similar position of tlieir directional ellipses. Similar position
includes identity of the angular directions mentioned above.
6. To reduce a given bivector r to the above form, Ave may
set
r.r = (cos q + i sin 1 ft].[a + ift]
= (cos 2 q -f i sin 2 q) (a.a—ft.ft)
— a+ ib
Avliere a and b are scalars, Avhicli we may regard as known.
The value of q may be determined by the equation
b
tan 2 q =
a
the quadrant to which 2 q belongs being determined so as to
give sin 2 q and cos 2 q the same signs as b and a. Then a and
ft will be given by the equation
ift — (cos q — i sin q) r.
The solution is indeterminate when the real and imaginary
parts of the given biA r ector are perpendicular and equal in
magnitude. In this case the directional ellipse is a circle, and
the bivector may be called circular. The criterion of a circular
bivector is
r.r = 0.
It is especially to be noticed that from this equation we can¬
not conclude that
r = 0,
as in the analysis of real vectors. This may also be shown by
expressing r in the form xi+yj+zA', in Avhich x, y, z are
biscalars. The equation then becomes
x 2 -\-y 2 -\-z 2 — 0,
which evidently does not require x, y, and s to vanish, as would
be the case if only real values are considered.
7. Def. —We call two vectors p and a perpendicular when
p,i r=0. Following the same analogy, Ave shall call two
bisectors v and § perpendicular, when
t.« = 0.
In considering the geometrical signification of this equation,
we shall first suppose that the real and imaginary components
of r and § lie in the same plane, and that both r and § have not
real directions. It is then evidently possible to express them
in the form
rn[a+ ift],
ift'\,
IT V ECTOR ANALYSIS.
81
Where m and m' are biscalar, a and $ are at right angles, and
a! parallel with /?. Then the equation r.8=0 requires that
D, and z=. 0.
This shows that the directional ellipses of the two bivectors are
similar and the angular direction from the real to the imag¬
inary component is the same in both, but the major axes of the
ellipses are perpendicular. The case in which the directions of
v and g are real, forms no exception to this rule.
It will be observed that every circular bivector is perpen¬
dicular to itself, and to every parallel bivector.
If two bivectors, y+tv, which do not lie in the same
plane are perpendicular, we may resolve y. and v into components
parallel and perpendicular to the plane of y! and v. The com¬
ponents perpendicular to the plane evidently contribute nothing
to the value of
[y+w ]. [p'-f iv']
Therefore the components of y and v parallel to the plane of //,
i/, form a bi vector which is perpendicular to fj! -V tv. That is,
if two bi vectors are perpendicular, the directional ellipse of
either, projected upon the plane of the other and rotated
through a quadrant in that plane, will be similar and similarly
situated to the directional ellipse of the second.
8. A bivector may be divided in one and only one way into
parts parallel and perpendicular to another, provided that the
second is not circular. If a and 6 are the bivectors, the parts
of a will be
b.a
U
6 aud a
b.a
b.b
b.
If b is circular, the resolution of a is impossible, unless it is
perpendicular to b. In this case the resolution is indeterminate.
9. Since axb.a—-0, and aXb.b = 0, axb is perpendicular to a
and b. We may regard the plane of the product as determined
by the condition that the directional ellipses of the factors pro¬
jected upon it become similar and similarly situated. The
directional ellipse of the product is similar to these projections,
but its orientation is different by 90°. It may easily be shown
that axb vanishes only with a or b, or when a and b are
parallel.
10. The bivector equation
(a X b. c) b — (b. c X b) a + (c. b X g) b — (b. a X b) c = 0
is identical, as may be verified by substituting expressions of
the form xi+yj+zfc, (x, y, z being bi scalars,) for each of the
bivectors. (Compare No. 37.) This equation shows that if the
82
BIVECTOR ANALYSIS.
product ctXb of any two bivectors vanishes, one of these will
be equal to the other with a biscalar coefficient, that is, they
will be parallel, according to the definition given above. If
the product ci.b X c of any three bivectors vanishes, the equation
shows that one of these may be expressed as a sum of the other
two with biscalar coefficients. In this case, we may say (from
the analogy of the scalar analysis) that the three bivectors are
complanar. (This does not imply that they lie in any same real
plane.) If ct.bXc is not equal to zero, the equation shows that
any fourth bivector may be expressed as a sum of a, b, and c
with biscalar coefficients, and indicates how these coefficients
may be determined.
11. The equation
(r.ci) b X c + (v.b) c X ft + (r.c) ft X b = (ft X b.c) r
is also identical, as may easily be verified. If we set
and suppose that
c = ci X b,
r.ci=0, r.b = 0,
the equation becomes
(v.ftXb) axb = (aXb.aXb) r.
This shows that if a bivector t is perpendicular to two bivectors
a and b, which are not parallel, r will be parallel to ctXb.
Therefore, all bivectors which are perpendicular to two given
hi vectors, are parallel to each other, unless the given two are
parallel.
BIVECTOR ANALYSIS,
83
ADDENDA ET CORRIGENDA.
Page 6, line 1 , for a=a + l3 read a=/3 + y.
Page 8, No. 33, change signs of third member of equation.
Page 11, line 7, after a.a'— (3.(M=y.y'=\. add as follows:
a.f3'= 0, *a.y'= 0, (3.a'= 0, [3.y'=Q, y.a'— 0, y.(3'=0.
These nine equations may be regarded as defining the relations between a, /?, y
and afV, y' as reciprocals.
Page 11, line 17, after y x a is add
a.' x [V, ft' x y', y' x a', or
Page 11, before No. 39, insert as follows:
38 a. If we multiply the identical equation (8) of No. 37 by a x r, we obtain the
equation
{a.(3 x y) ( p.(T x r)= a.p({3ya y. t—i3.t y.a)
-4- j3.p(y.a a.T—y.r a.a) -f y.p(a.a [J.t — a.r (3. a ),
which is therefore identical. But this equation cannot subsist identically, unless
(a.j3 x y)a x T=a(j3.a y.T—j3.r y.a ) + (3(y.a a.T—y.r a. a) 4- y(a.a (3.r — a.r f3.o)
is also an identical equation. (The reader will observe that in each of these
equations the second member may be expressed as a determinant.)
Prom these transformations, with those already given, it follows that a product
formed of any number of letters (representing vectors and scalars), combined in
any possible way by scalar, direct, and skew multiplications, may be reduced to
a sum of products, containing each the sign x once and only once, when the
original product contains it an odd number of times, or entirely free from the
sign, when the original product contains it an even number of times.
Page 15, line 7 from foot, in denominator of fraction,
for
dp dp
ds ’ ds
read
d' 2 p d' 2 p
ds' 2 ' ds' 2 '
Page 18, line 10 from foot, after continuous add and single-valued.
Page 27, line 6, for u— constant read t—u= constant.
Page 29, line 4, for ydu read y.Ow.
Page 33, for second and third lines of No. 98, read
V-V Pot« = v-V Pot \ui 3 -vj + ulc\
— v-Y Pot u % 4- v-V P°t V J + V-V Pot A*
= — 4t xu % — 4 fcvj — 47TW Jc.
= — 47ry.
Page 36, line 5 from foot, after u, read aud that in the shell (compare
No. 90).
Page 36, line 4 from foot, dele (No. 90).
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